(a)

\begin{align} x^2 - 7x - 8 & = 0 \\ (x + 1)(x - 8) & = 0 \end{align} \begin{align} x + 1 & = 0 && \text{ or } & x - 8 & = 0 \\ x & = -1 &&& x & = 8 \end{align}

(b)

\begin{align} 2y^2 + 3y - 20 & = 0 \\ (y + 4)(2y - 5) & = 0 \end{align} \begin{align} y + 4 & = 0 && \text{ or } & 2y - 5 & = 0 \\ y & = -4 &&& 2y & = 5 \\ & &&& y & = {5 \over 2} \end{align}

(a)

\begin{align} (x + 7)^2 & = 100 \\ x + 7 & = \pm \sqrt{100} \\ x + 7 & = \pm 10 \end{align} \begin{align} x + 7 & = 10 && \text{ or } & x + 7 & = -10 \\ x & = 10 - 7 &&& x & = -10 - 7 \\ x & = 3 &&& x & = -17 \end{align}

(b)

\begin{align} (2y - 5)^2 & = 11 \\ 2y - 5 & = \pm \sqrt{11} \end{align} \begin{align} 2y - 5 & = \sqrt{11} && \text{ or } & 2y - 5 & = - \sqrt{11} \\ 2y & = \sqrt{11} + 5 &&& 2y & = - \sqrt{11} + 5 \\ y & = {\sqrt{11} + 5 \over 2} &&& y & = {-\sqrt{11} + 5 \over 2} \\ y & \approx 4.158312 &&& y & \approx 0.841688 \\ y & \approx 4.16 &&& y & \approx 0.842 \end{align}

(a)

\begin{align} x^2 + 12x & = x^2 + 12x + \left(12 \over 2\right)^2 - \left(12 \over 2\right)^2 \\ & = x^2 + 12x + 6^2 - 6^2 \\ & = (x + 6)^2 - 36 \end{align}

(b)

\begin{align} x^2 - 7x & = x^2 - 7x + \left(7 \over 2\right)^2 - \left(7 \over 2\right)^2 \\ & = \left(x - {7 \over 2}\right)^2 - \left(7 \over 2\right)^2 \\ & = \left(x - {7 \over 2}\right)^2 - {49 \over 4} \end{align}

(c)

\begin{align} x^2 + 1.6x & = x^2 + 1.6x + \left(1.6 \over 2\right)^2 - \left(1.6 \over 2\right)^2 \\ & = x^2 + 1.6x + (0.8)^2 - (0.8)^2 \\ & = (x + 0.8)^2 - (0.8)^2 \\ & = (x + 0.8)^2 - 0.64 \end{align}

(d)

\begin{align} x^2 - {3 \over 4}x & = x^2 - {3 \over 4}x + \left( {3 \over 4} \over 2 \right)^2 - \left( {3 \over 4} \over 2 \right)^2 \\ & = x^2 - {3 \over 4}x + \left(3 \over 8\right)^2 - \left(3 \over 8\right)^2 \\ & = \left(x - {3 \over 8}\right)^2 - \left(3 \over 8\right)^2 \\ & = \left(x - {3 \over 8}\right)^2 - {9 \over 64} \end{align}

(a)

\begin{align} x^2 + 14x + 5 & = x^2 + 14x + \left(14 \over 2\right)^2 - \left(14 \over 2\right)^2 + 5 \\ & = x^2 + 14x + 7^2 - 7^2 + 5 \\ & = (x + 7)^2 - 7^2 + 5 \\ & = (x + 7)^2 - 44 \end{align}

(b)

\begin{align} x^2 + 7x - 1.2 & = x^2 + 7x + \left(7 \over 2\right)^2 - \left(7 \over 2\right)^2 - 1.2 \\ & = x^2 + 7x + (3.5)^2 - (3.5)^2 - 1.2 \\ & = (x + 3.5)^2 - (3.5)^2 - 1.2 \\ & = (x + 3.5)^2 - 13.45 \end{align}

(c)

\begin{align} x^2 - 9x + 3 & = x^2 - 9x + \left(9 \over 2\right)^2 - \left(9 \over 2\right)^2 + 3 \\ & = \left(x - {9 \over 2}\right)^2 - \left(9 \over 2\right)^2 + 3 \\ & = \left(x - {9 \over 2}\right)^2 - {69 \over 4} \end{align}

(d)

\begin{align} x^2 - {6 \over 5}x - 4 & = x^2 - {6 \over 5}x + \left({6 \over 5} \over 2\right)^2 - \left({6 \over 5} \over 2\right)^2 - 4 \\ & = x^2 - {6 \over 5}x + \left(3 \over 5\right)^2 - \left(3 \over 5\right)^2 - 4 \\ & = \left(x - {3 \over 5}\right)^2 - \left(3 \over 5\right)^2 - 4 \\ & = \left(x - {3 \over 5}\right)^2 - {109 \over 25} \end{align}

(a)

\begin{align} -x^2 + 6x - 2 & = -(x^2 - 6x) - 2 \\ & = - \left[ x^2 - 6x + \left(6 \over 2\right)^2 - \left(6 \over 2\right)^2 \right] - 2 \\ & = - (x^2 - 6x + 3^2 - 3^2) - 2 \\ & = - [(x - 3)^2 - 3^2] - 2 \\ & = - [(x - 3)^2 - 9] - 2 \\ & = - (x - 3)^2 + 9 - 2 \\ & = - (x - 3)^2 + 7 \end{align}

(b)

\begin{align} - x^2 + 9x - 3.5 & = -(x^2 - 9x) - 3.5 \\ & = - \left[ x^2 - 9x + \left(9 \over 2\right)^2 - \left(9 \over 2\right)^2 \right] - 3.5 \\ & = - ( x^2 - 9x + 4.5^2 - 4.5^2 ) - 3.5 \\ & = - [ (x - 4.5)^2 - 4.5^2 ] - 3.5 \\ & = - [ (x - 4.5)^2 - 20.25 ] - 3.5 \\ & = - (x - 4.5)^2 + 20.25 - 3.5 \\ & = - (x - 4.5)^2 + 16.75 \end{align}

(c)

\begin{align} - x^2 - 7x + 5 & = -(x^2 + 7x) + 5 \\ & = - \left[ x^2 + 7x + \left(7 \over 2\right)^2 - \left(7 \over 2\right)^2 \right] + 5 \\ & = - \left[ \left(x + {7 \over 2}\right)^2 - \left(7 \over 2\right)^2 \right] + 5 \\ & = - \left[ \left(x + {7 \over 2}\right)^2 - {49 \over 4} \right] + 5 \\ & = - \left(x + {7 \over 2}\right)^2 + {49 \over 4} + 5 \\ & = - \left(x + {7 \over 2}\right)^2 + {69 \over 4} \end{align}

(d)

\begin{align} - x^2 - {4 \over 9}x - 1 & = - \left(x^2 + {4 \over 9}x\right) - 1 \\ & = - \left[ x^2 + {4 \over 9}x + \left({4 \over 9} \over 2\right)^2 - \left({4 \over 9} \over 2\right)^2 \right] - 1 \\ & = - \left[ x^2 + {4 \over 9}x + \left({2 \over 9}\right)^2 - \left({2 \over 9}\right)^2 \right] - 1 \\ & = - \left[ \left(x + {2 \over 9}\right)^2 - \left(2 \over 9\right)^2 \right] - 1 \\ & = - \left[ \left(x + {2 \over 9}\right)^2 - {4 \over 81} \right] - 1 \\ & = - \left(x + {2 \over 9}\right)^2 + {4 \over 81} - 1 \\ & = - \left(x + {2 \over 9}\right)^2 - {77 \over 81} \end{align}

(a)

\begin{align} x^2 + 6x - 4 & = 0 \\ x^2 + 6x + \left(6 \over 2\right)^2 - \left(6 \over 2\right)^2 - 4 & = 0 \\ x^2 + 6x + 3^2 - 3^2 - 4 & = 0 \\ (x + 3)^2 - 3^2 - 4 & = 0 \\ (x + 3)^2 - 13 & = 0 \\ (x + 3)^2 & = 13 \\ x + 3 & = \pm \sqrt{13} \end{align} \begin{align} x + 3 & = \sqrt{13} && \text{ or } & x + 3 & = - \sqrt{13} \\ x & = \sqrt{13} - 3 &&& x & = - \sqrt{13} - 3 \\ x & \approx 0.605551 &&& x & \approx -6.605551 \\ x & \approx 0.61 \text{ (2 d.p.)} &&& x & \approx -6.61 \text{ (2 d.p.)} \end{align}

(b)

\begin{align} y^2 + 7y + 5 & = 0 \\ y^2 + 7y + \left(7 \over 2\right)^2 - \left(7 \over 2\right)^2 + 5 & = 0 \\ y^2 + 7y + 3.5^2 - 3.5^2 + 5 & = 0 \\ (y + 3.5)^2 - 3.5^2 + 5 & = 0 \\ (y + 3.5)^2 - 7.25 & = 0 \\ (y + 3.5)^2 & = 7.25 \\ y + 3.5 & = \pm \sqrt{7.25} \end{align} \begin{align} y + 3.5 & = \sqrt{7.25} && \text{ or } & y + 3.5 & = -\sqrt{7.25} \\ y & = \sqrt{7.25} - 3.5 &&& y & = - \sqrt{7.25} - 3.5 \\ y & \approx -0.807418 &&& y & \approx -6.192582 \\ y & \approx -0.81 \text{ (2 d.p.)} &&& y & \approx -6.19 \text{ (2 d.p.)} \end{align}

(c)

\begin{align} z^2 - z - 1 & = 0 \\ z^2 - z + \left(1 \over 2\right)^2 - \left(1 \over 2\right)^2 - 1 & = 0 \\ z^2 - z + 0.5^2 - 0.5^2 - 1 & = 0 \\ (z - 0.5)^2 - 0.5^2 - 1 & = 0 \\ (z - 0.5)^2 - 1.25 & = 0 \\ (z - 0.5)^2 & = 1.25 \\ z - 0.5 & = \pm \sqrt{1.25} \end{align} \begin{align} z - 0.5 & = \sqrt{1.25} && \text{ or } & z - 0.5 & = -\sqrt{1.25} \\ z & = \sqrt{1.25} + 0.5 &&& z & = - \sqrt{1.25} + 0.5 \\ z & \approx 1.618034 &&& z & \approx -0.618034 \\ z & \approx 1.62 \text{ (2 d.p.)} &&& z & \approx -0.62 \text{ (2 d.p.)} \end{align}

\begin{align} (x + 4)(x - 3) & = 15 \\ x^2 - 3x + 4x - 12 & = 15 \\ x^2 + x - 12 & = 15 \\ x^2 + x - 12 - 15 & = 0 \\ x^2 + x - 27 & = 0 \\ x^2 + x + \left(1 \over 2\right)^2 - \left(1 \over 2\right)^2 - 27 & = 0 \\ x^2 + x + 0.5^2 - 0.5^2 - 27 & = 0 \\ (x + 0.5)^2 - 0.5^2 - 27 & = 0 \\ (x + 0.5)^2 - 27.25 & = 0 \\ (x + 0.5)^2 & = 27.25 \\ x + 0.5 & = \pm \sqrt{27.25} \end{align} \begin{align} x + 0.5 & = \sqrt{27.25} && \text{ or } & x + 0.5 & = - \sqrt{27.25} \\ x & = \sqrt{27.25} - 0.5 &&& x & = - \sqrt{27.25} - 0.5 \\ x & \approx 4.720153 &&& x & \approx -5.720153 \\ x & \approx 4.72 &&& x & \approx -5.72 \end{align}

(a)

\begin{align} 2x^2 & + 3x - 7 = 0 \\ \\ x & = {- b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 3 \pm \sqrt{(3)^2 - 4(2)(-7)} \over 2(2)} \\ & = {-3 \pm \sqrt{65} \over 4} \\ & \approx 1.265564 \text{ or } -2.765564 \\ & \approx 1.27 \text{ or } -2.77 \end{align}

(b)

\begin{align} -5x^2 & + 8x + 1 = 0 \\ \\ x & = {- b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 8 \pm \sqrt{(8)^2 - 4(-5)(1)} \over 2(-5)} \\ & = {- 8 \pm \sqrt{84} \over -10} \\ & \approx -0.116515 \text{ or } 1.716515 \\ & \approx -0.117 \text{ or } 1.72 \end{align}

(c)

\begin{align} 3x^2 & - 5 - x = 0 \\ 3x^2 & - x - 5 = 0 \\ \\ x & = {- b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-1) \pm \sqrt{(-1)^2 - 4(3)(-5)} \over 2(3)} \\ & = {1 \pm \sqrt{61} \over 6} \\ & \approx 1.468375 \text{ or } -1.135042 \\ & \approx 1.47 \text{ or } -1.14 \end{align}

(d)

\begin{align} 1 & - x^2 - 7x = 0 \\ -x^2 &- 7x + 1 \phantom{(} = 0 \\ \\ x & = {- b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-7) \pm \sqrt{(-7)^2 - 4(-1)(1)} \over 2(-1)} \\ & = {7 \pm \sqrt{53} \over -2} \\ & \approx -7.140055 \text{ or } 0.140055 \\ & \approx -7.14 \text{ or } 0.140 \end{align}

(e)

\begin{align} (x - 1)^2 & = 4x - 5 \\ (x)^2 - 2(x)(1) + (1)^2 & = 4x - 5 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ x^2 - 2x + 1 & = 4x - 5 \\ x^2 - 2x - 4x + 1 + 5 & = 0 \\ x^2 - 6x + 6 & = 0 \\ \\ \\ x & = {- b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-6) \pm \sqrt{(-6)^2 - 4(1)(6)} \over 2(1)} \\ & = {6 \pm \sqrt{12} \over 2} \\ & \approx 4.732051 \text{ or } 1.267949 \\ & \approx 4.73 \text{ or } 1.27 \end{align}

(f)

\begin{align} (x + 3)(x - 1) & = 8x - 7 \\ x^2 - x + 3x - 3 & = 8x - 7 \\ x^2 + 2x - 3 & = 8x - 7 \\ x^2 + 2x - 8x - 3 + 7 & = 0 \\ x^2 - 6x + 4 & = 0 \\ \\ \\ x & = {- b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-6) \pm \sqrt{(-6)^2 - 4(1)(4)} \over 2(1)} \\ & = {6 \pm \sqrt{20} \over 2} \\ & \approx 5.236068 \text{ or } 0.763932 \\ & \approx 5.24 \text{ or } 0.764 \end{align}

(a)

\begin{align} {4 \over p} & = 2p - 3 \\ {4 \over p} & = {2p - 3 \over 1} \\ 4 & = p(2p - 3) \\ 4 & = 2p^2 - 3p \\ 0 & = 2p^2 - 3p - 4 \\ \\ \\ p & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-3) \pm \sqrt{(-3)^2 - 4(2)(-4)} \over 2(2) } \\ & = {3 \pm \sqrt{41} \over 4} \\ & \approx 2.350781 \text{ or } -0.850781 \\ & \approx 2.35 \text{ or } -0.851 \end{align}

(b)

\begin{align} {2q \over 10 - 3q} & = 10 - 3q \\ {2q \over 10 - 3q} & = {10 - 3q \over 1} \\ 2q & = (10 - 3q)(10 - 3q) \\ 2q & = 100 - 30q - 30q + 9q^2 \\ 2q & = 9q^2 - 60q + 100 \\ 0 & = 9q^2 - 60q - 2q + 100 \\ 0 & = 9q^2 - 62q + 100 \\ \\ \\ q & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-62) \pm \sqrt{(-62)^2 - 4(9)(100)} \over 2(9) } \\ & = {62 \pm \sqrt{244} \over 18} \\ & \approx 4.312250 \text{ or } 2.576639 \\ & \approx 4.31 \text{ or } 2.58 \end{align}

(c)

\begin{align} {6 \over x + 4} & = x + 3 \\ {6 \over x + 4} & = {x + 3 \over 1} \\ 6 & = (x + 4)(x + 3) \\ 6 & = x^2 + 3x + 4x + 12 \\ 6 & = x^2 + 7x + 12 \\ 0 & = x^2 + 7x + 12 - 6 \\ 0 & = x^2 + 7x + 6 \\ 0 & = (x + 1)(x + 6) \end{align} \begin{align} x + 1 & = 0 && \text{ or } & x + 6 & = 0 \\ x & = - 1 &&& x & = -6 \end{align}

(d)

\begin{align} {3 \over 12 - y} & = 3y - 1 \\ {3 \over 12 - y} & = {3y - 1 \over 1} \\ 3 & = (12 - y)(3y - 1) \\ 3 & = 36y - 12 - 3y^2 + y \\ 3 & = - 3y^2 + 37y - 12 \\ 0 & = - 3y^2 + 37y - 12 - 3 \\ 0 & = - 3y^2 + 37y - 15 \\ \\ \\ y & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 37 \pm \sqrt{(37)^2 - 4(-3)(-15)} \over 2(-3) } \\ & = {- 37 \pm \sqrt{1189} \over -6} \\ & \approx 0.419687 \text{ or } 11.913647 \\ & \approx 0.420 \text{ or } 11.9 \end{align}

(a)

\begin{align} {1 \over x + 6} + {2 \over 3 - x} & = 5 \\ {3 - x \over (x + 6)(3 - x)} + {2(x + 6) \over (x + 6)(3 - x)} & = 5 \\ {3 - x + 2(x + 6) \over (x + 6)(3 - x) } & = 5 \\ {3 - x + 2x + 12 \over (x + 6)(3 - x)} & = 5 \\ {x + 15 \over (x + 6)(3 - x)} & = {5 \over 1} \\ x + 15 & = 5(x + 6)(3 - x) \\ x + 15 & = 5(3x - x^2 + 18 - 6x) \\ x + 15 & = 5(-x^2 - 3x + 18) \\ x + 15 & = - 5x^2 - 15x + 90 \\ 0 & = - 5x^2 - 15x - x + 90 - 15 \\ 0 & = - 5x^2 - 16x + 75 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-16) \pm \sqrt{(-16)^2 - 4(-5)(75)} \over 2(-5) } \\ & = { 16 \pm \sqrt{1756} \over -10} \\ & \approx -5.790465 \text{ or } 2.590465 \\ & \approx -5.79 \text{ or } 2.59 \end{align}

(b)

\begin{align} {5 \over y - 3} + {y - 1 \over y - 2} & = 7 \\ {5(y - 2) \over (y - 3)(y - 2)} + {(y - 1)(y - 3) \over (y - 3)(y - 2)} & = 7 \\ {5(y - 2) + (y - 1)(y - 3) \over (y - 3)(y - 2)} & = 7 \\ {5y - 10 + y^2 - 3y - y + 3 \over y^2 - 2y - 3y + 6} & = 7 \\ {y^2 + y - 7 \over y^2 - 5y + 6} & = {7 \over 1} \\ y^2 + y - 7 & = 7(y^2 - 5y + 6) \\ y^2 + y - 7 & = 7y^2 - 35y + 42 \\ 0 & = 7y^2 - y^2 - 35y - y + 42 + 7 \\ 0 & = 6y^2 - 36y + 49 \\ \\ \\ y & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-36) \pm \sqrt{(-36)^2 - 4(6)(49)} \over 2(6) } \\ & = { 36 \pm \sqrt{120} \over 12} \\ & \approx 3.912871 \text{ or } 2.087129 \\ & \approx 3.91 \text{ or } 2.09 \end{align}

(c)

\begin{align} {3 \over n - 2} - {1 \over (n - 2)^2} & = -2 \\ {3(n - 2) \over (n - 2)^2} - {1 \over (n - 2)^2} & = -2 \\ {3(n - 2) - 1 \over (n - 2)^2} & = -2 \\ {3n - 6 - 1 \over (n)^2 - 2(n)(2) + (2)^2} & = -2 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ {3n - 7 \over n^2 - 4n + 4} & = {-2 \over 1} \\ 3n - 7 & = -2(n^2 - 4n + 4) \\ 3n - 7 & = - 2n^2 + 8n - 8 \\ 0 & = -2n^2 + 8n - 3n - 8 + 7 \\ 0 & = -2n^2 + 5n - 1 \\ \\ \\ n & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 5 \pm \sqrt{(5)^2 - 4(-2)(-1)} \over 2(-2) } \\ & = { -5 \pm \sqrt{17} \over -4} \\ & \approx 0.219224 \text{ or } 2.280776 \\ & \approx 0.219 \text{ or } 2.28 \end{align}

(d)

\begin{align} {6 \over 1 - 2t} + {3t \over 1 - 4t^2} & = 1 \\ {6 \over 1 - 2t} + {3t \over 1^2 - (2t)^2} & = 1 \\ {6 \over 1 - 2t} + {3t \over (1 - 2t)(1 + 2t)} & = 1 \phantom{000000} [a^2 - b^2 = (a - b)(a + b)] \\ {6(1 + 2t) \over (1 - 2t)(1 + 2t)} + {3t \over (1 - 2t)(1 + 2t)} & = 1 \\ {6(1 + 2t) + 3t \over (1 - 2t)(1 + 2t)} & = 1 \\ {6 + 12t + 3t \over (1 - 2t)(1 + 2t)} & = 1 \\ {6 + 15t \over (1 - 2t)(1 + 2t)} & = {1 \over 1} \\ 6 + 15t & = (1 - 2t)(1 + 2t) \\ 6 + 15t & = 1 - 4t^2 \\ 4t^2 + 15t + 6 - 1 & = 0 \\ 4t^2 + 15t + 5 & = 0 \\ \\ \\ t & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 15 \pm \sqrt{(15)^2 - 4(4)(5)} \over 2(4) } \\ & = { - 15 \pm \sqrt{145} \over 8} \\ & \approx -0.369801 \text{ or } -3.380199 \\ & \approx -0.370 \text{ or } -3.38 \end{align}

(i)

(ii)

(iii) Look for the x-intercepts of the graph

\begin{align} \text{From graph, } & x = -0.2 \text{ or } 2.2 \end{align}

(i)

x	-1	0	1	2	3	4	5	6
y	16	9	4	1	0	1	4	9

(ii) Look for the x-intercept(s) of the graph

\begin{align} \text{From graph, } & x = 3 \end{align}

(a)

\begin{align} y & = (x - 2)(x - 6) \\ y & = x^2 - 6x - 2x + 12 \\ y & = x^2 - 8x + 12 \phantom{000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (0 - 2)(0 - 6) \\ y & = 12 \phantom{00000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x - 2)(x - 6) \\ \\ x & = 2 \text{ or } x = 6 \phantom{00000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {2 + 6 \over 2} \\ x & = 4 \\ \\ \text{Let } & x = 4, \\ y & = (4 - 2)(4 - 6) \\ y & = -4 \\ \\ \text{Minimum point: } & (4, -4) \end{align}

(b)

\begin{align} y & = (x + 4)(x + 7) \\ y & = x^2 + 7x + 4x + 28 \\ y & = x^2 + 11x + 28 \phantom{000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (0 + 4)(0 + 7) \\ y & = 28 \phantom{000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x + 4)(x + 7) \\ \\ x & = -4 \text{ or } x = -7 \phantom{0000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {-4 + (-7) \over 2} \\ x & = -5.5 \\ \\ \text{Let } & x = -5.5, \\ y & = (-5.5 + 4)(-5.5 + 7) \\ y & = -2.25 \\ \\ \text{Minimum point: } & (-5.5, -2.25) \end{align}

(c)

\begin{align} y & = -x(x - 5) \\ y & = - x^2 + 5x \phantom{00000000.} [\text{Maximum curve } (\cap)] \\ \\ \text{Let } & x = 0, \\ y & = -(0)(0 - 5) \\ y & = 0 \phantom{000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = -x(x - 5) \\ \\ x & = 0 \text{ or } x = 5 \phantom{00000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {0 + 5 \over 2} \\ x & = 2.5 \\ \\ \text{Let } & x = 2.5, \\ y & = -(2.5)(2.5 - 5) \\ y & = 6.25 \\ \\ \text{Maximum point: } & (2.5, 6.25) \end{align}

(d)

\begin{align} y & = (3 - x)(x + 8) \\ y & = 3x + 24 - x^2 - 8x \\ y & = - x^2 - 5x + 24 \phantom{00000000.} [\text{Maximum curve } (\cap)] \\ \\ \text{Let } & x = 0, \\ y & = (3 - 0)(0 + 8) \\ y & = 24 \phantom{000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (3 - x)(x + 8) \\ \\ x & = 3 \text{ or } x = -8 \phantom{0000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {3 + (-8) \over 2} \\ x & = -2.5 \\ \\ \text{Let } & x = -2.5, \\ y & = [3 - (-2.5)](-2.5 + 8) \\ y & = 30.25 \\ \\ \text{Maximum point: } & (-2.5, 30.25) \end{align}

(i)

\begin{align} y & = -x^2 + 4x - 6 \\ & = - (x^2 - 4x) - 6 \\ & = - \left[ x^2 - 4x + \left(4 \over 2\right)^2 - \left(4 \over 2\right)^2 \right] - 6 \phantom{000000} [\text{Complete the square}] \\ & = - (x^2 - 4x + 2^2 - 2^2) - 6 \\ & = - [ (x - 2)^2 - 4 ] - 6 \\ & = - (x - 2)^2 + 4 - 6 \\ & = - (x - 2)^2 - 2 \end{align}

(ii)

\begin{align} y & = -x^2 + 4x - 6 \\ y & = -(x - 2)^2 - 2 \phantom{00000000.} [\text{Maximum curve } (\cap)] \\ \\ \text{Maximum point: } & (2, -2) \\ \\ \text{Let } & x = 0, \\ y & = - (0 - 2)^2 - 2 \\ y & = -6 \phantom{00000000000000000} [y \text{-intercept}] \end{align}

(i)

\begin{align} y & = x^2 + x + 1 \\ & = x^2 + x + \left(1 \over 2\right)^2 - \left(1 \over 2\right)^2 + 1 \phantom{000000} [\text{Complete the square}] \\ & = x^2 + x + (0.5)^2 - (0.5)^2 + 1 \\ & = (x + 0.5)^2 - 0.25 + 1 \\ & = (x + 0.5)^2 + 0.75 \end{align}

(ii)

\begin{align} y & = x^2 + x + 1 \\ y & = (x + 0.5)^2 + 0.75 \\ y & = [x - (-0.5)]^2 + 0.75 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Minimum point: } & (-0.5, 0.75) \\ \\ \text{Let } & x = 0, \\ y & = (0 + 0.5)^2 + 0.75 \\ y & = 1 \phantom{000000000000000000000} [y \text{-intercept}] \end{align}

(i)

\begin{align} y & = - x^2 - 6x - 9 \\ & = -(x^2 + 6x + 9) \\ & = - \left[ x^2 + 6x + \left(6 \over 2\right)^2 - \left(6 \over 2\right)^2 + 9 \right] \phantom{000000} [\text{Complete the square}] \\ & = - (x^2 + 6x + 3^2 - 3^2 + 9) \\ & = - [ (x + 3)^2 - 9 + 9 ] \\ & = - [(x + 3)^2 ] \\ & = - (x + 3)^2 \end{align}

(ii)

\begin{align} y & = -x^2 - 6x - 9 \\ y & = -(x + 3)^2 \\ y & = -[x - (-3)]^2 + 0 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ \\ \text{Maximum point: } & (-3, 0) \\ \\ \text{Let } & x = 0, \\ y & = -(0 + 3)^2 \\ y & = -9 \phantom{000000000000000000000} [y \text{-intercept}] \end{align}

(i)

\begin{align} \text{Perimeter} & = AC + BC + AB \\ 17 & = 8 + x + AB \\ \\ AB & = 17 - 8 - x \\ & = (9 - x) \text{ m} \end{align}

(ii)

\begin{align} \text{By Pyth} & \text{agoras theorem,} \\ AC^2 & = AB^2 + BC^2 \\ (8)^2 & = (9 - x)^2 + (x)^2 \\ 64 & = \underbrace{(9)^2 - 2(9)(x) + (x)^2}_{(a - b)^2 = a^2 - 2ab + b^2} + x^2 \\ 64 & = 81 - 18x + x^2 + x^2 \\ 64 & = 2x^2 - 18x + 81 \\ 0 & = 2x^2 - 18x + 81 - 64 \\ 0 & = 2x^2 - 18x + 17 \phantom{00} \text{ (Shown)} \end{align}

(iii)

\begin{align} 0 & = 2x^2 - 18x + 17 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-18) \pm \sqrt{(-18)^2 - 4(2)(17)} \over 2(2)} \\ & = {18 \pm \sqrt{188} \over 4} \\ & \approx 7.927827 \text{ or } 1.072173 \\ & \approx 7.928 \text{ or } 1.072 \text{ (3 d.p.)} \end{align}

(iv)

\begin{align} \text{Since } AB \text{ is longer than } BC, \phantom{.} x \approx 1.072173 \text{ is the only possible solution} \end{align}

(v)

\begin{align} \text{From (ii), } 2x^2 - 18x + 17 & = 0 \\ 2x(9 - x) & = 17 \\ x(9 - x) & = {17 \over 2} \\ \\ \text{Volume of prism} & = \text{Cross-sectional area} \times \text{Height} \\ & = {1 \over 2} \times AB \times BC \times 7 \\ & = {1 \over 2} \times (9 - x) \times x \times 7 \\ & = {1 \over 2} \times {17 \over 2} \times 7 \\ & = 29.75 \text{ m}^3 \\ & \approx 29.8 \text{ m}^3 \text{ (3 s.f.)} \end{align}

(i)

\begin{align} \text{Time} & = { \text{Distance} \over \text{Speed} } \\ \\ \text{Time taken for initial journey} & = { 600 \over x} \text{ hours} \\ \\ \text{Time taken for return journey} & = {600 \over x + 7} \text{ hours} \\ \\ 15 \text{ mins} & = {15 \over 60} \\ & = {1 \over 4} \text{ hours} \\ \\ {600 \over x} - {600 \over x + 7} & = {1 \over 4} \\ {600(x + 7) \over x(x + 7)} - {600x \over x(x + 7)} & = {1 \over 4} \\ {600(x + 7) - 600x \over x(x + 7)} & = {1 \over 4} \\ {600x + 4200 - 600x \over x^2 + 7x} & = {1 \over 4} \\ {4200 \over x^2 + 7x} & = {1 \over 4} \\ 4(4200) & = x^2 + 7x \\ 16 \phantom{.} 800 & = x^2 + 7x \\ 0 & = x^2 + 7x - 16 \phantom{.} 800 \phantom{00} \text{ (Shown)} \end{align}

(ii)

\begin{align} 0 & = x^2 + 7x - 16 \phantom{.} 800 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-7 \pm \sqrt{(7)^2 - 4(1)(-16 \phantom{.} 800)} \over 2(1)} \\ & = {-7 \pm \sqrt{67 \phantom{.} 249} \over 2} \\ & \approx 126.162061 \text{ or } -133.162061 \\ & \approx 126.16 \text{ or } -133.16 \text{ ( 2 d.p.)} \end{align}

(iii)

\begin{align} \text{From (i), time taken for return journey} & = {600 \over x + 7} \\ & = {600 \over {-7 + \sqrt{67249} \over 2} + 7} \\ & \approx 4.505788 \text{ hours} \\ & \approx 4 \text{ hours } 30.347273 \text{ mins} \\ & \approx 4 \text{ hours } 30 \text{ mins} \end{align}

\begin{align} y & = -x^2 + 2x + 0.5 \\ \\ \text{When } & x = 1, \phantom{000000000000000} [\text{Platform is 1 m away from bell}] \\ y & = -(1)^2 + 2(1) + 0.5 \\ y & = 1.5 \\ \\ \implies \text{Vani's } & \text{feet is 1.5 m from the ground} \\ \\ \text{Height Vani can reach} & = 1.5 + 1.6 \\ & = 3.1 \text{ m} \\ \\ \therefore \text{Vani } & \text{can reach the bell} \end{align}

(i)

(ii)

\begin{align} \text{From graph, } x & = 7.5 \end{align}

(a)

\begin{align} x^2 + 7x - 18 & = 0 \\ (x + 9)(x - 2) & = 0 \end{align} \begin{align} x + 9 & = 0 && \text{ or } & x - 2 & = 0 \\ x & = - 9 &&& x & = 2 \end{align}

(b)

\begin{align} 2x^2 + 5x - 7 & = 0 \\ (2x + 7)(x - 1) & = 0 \end{align} \begin{align} 2x + 7 & = 0 && \text{ or } & x - 1 & = 0 \\ 2x & = -7 &&& x & = 1 \\ x & = -{7 \over 2} \end{align}

(c)

\begin{align} 5y^2 - 28y + 15 & = 0 \\ (5y - 3)(y - 5) & = 0 \end{align} \begin{align} 5y - 3 & = 0 && \text{ or } & y - 5 & = 0 \\ 5y & = 3 &&& y & = 5 \\ y & = {3 \over 5} \end{align}

(d)

\begin{align} 4z^2 - 49 & = 0 \\ 4z^2 & = 49 \\ z^2 & = {49 \over 4} \\ z & = \pm \sqrt{49 \over 4} \\ z & = \pm {7 \over 2} \end{align}

(a)

\begin{align} (x + 1)^2 & = 9 \\ x + 1 & = \pm \sqrt{9} \\ x + 1 & = \pm 3 \end{align} \begin{align} x + 1 & = 3 && \text{ or } & x + 1 & = -3 \\ x & = 3 - 1 &&& x & = -3 - 1 \\ x & = 2 &&& x & = -4 \end{align}

(b)

\begin{align} (2y + 1)^2 & = 16 \\ 2y + 1 & = \pm \sqrt{16} \\ 2y + 1 & = \pm 4 \end{align} \begin{align} 2y + 1 & = 4 && \text{ or } & 2y + 1 & = -4 \\ 2y & = 4 - 1 &&& 2y & = -4 - 1 \\ 2y & = 3 &&& 2y & = -5 \\ y & = {3 \over 2} &&& y & = -{5 \over 2} \end{align}

(c)

\begin{align} (5h - 4)^2 & = 81 \\ 5h - 4 & = \pm \sqrt{81} \\ 5h - 4 & = \pm 9 \end{align} \begin{align} 5h - 4 & = 9 && \text{ or } & 5h - 4 & = -9 \\ 5h & = 9 + 4 &&& 5h & = -9 + 4 \\ 5h & = 13 &&& 5h & = -5 \\ h & = {13 \over 5} &&& h & = {-5 \over 5} \\ & &&& & = -1 \end{align}

(d)

\begin{align} (7 - 3k)^2 & = {9 \over 16} \\ 7 - 3k & = \pm \sqrt{9 \over 16} \\ 7 - 3k & = \pm {3 \over 4} \end{align} \begin{align} 7 - 3k & = {3 \over 4} && \text{ or } & 7 - 3k & = -{3 \over 4} \\ -3k & = {3 \over 4} - 7 &&& -3k & = -{3 \over 4} - 7 \\ -3k & = -{25 \over 4} &&& -3k & = -{31 \over 4} \\ k & = {-{25 \over 4} \over -3} &&& k & = {-{31 \over 4} \over -3} \\ k & = {25 \over 12} &&& k & = {31 \over 12} \end{align}

(e)

\begin{align} (m + 3)^2 & = 11 \\ m + 3 & = \pm \sqrt{11} \end{align} \begin{align} m + 3 & = \sqrt{11} && \text{ or } & m + 3 & = - \sqrt{11} \\ m & = \sqrt{11} - 3 &&& m & = - \sqrt{11} - 3 \\ m & \approx 0.31662 &&& m & \approx -6.3166 \\ m & \approx 0.32 \text{ (2 d.p.)} &&& m & \approx -6.32 \text{ (2 d.p.)} \end{align}

(f)

\begin{align} (2n - 3)^2 & = 23 \\ 2n - 3 & = \pm \sqrt{23} \end{align} \begin{align} 2n - 3 & = \sqrt{23} && \text{ or } & 2n - 3 & = - \sqrt{23} \\ 2n & = \sqrt{23} + 3 &&& 2n & = - \sqrt{23} + 3 \\ n & = {\sqrt{23} + 3 \over 2} &&& n & = {-\sqrt{23} + 3 \over 2} \\ n & \approx 3.8979 &&& n & \approx -0.89792 \\ n & \approx 3.90 \text{ (2 d.p.)} &&& n & \approx -0.90 \text{ (2 d.p.)} \end{align}

(g)

\begin{align} (5 - w)^2 & = 7 \\ 5 - w & = \pm \sqrt{7} \end{align} \begin{align} 5 - w & = \sqrt{7} && \text{ or } & 5 - w & = -\sqrt{7} \\ -w & = \sqrt{7} - 5 &&& -w & = - \sqrt{7} - 5 \\ w & = -\sqrt{7} + 5 &&& w & = \sqrt{7} + 5 \\ w & \approx 2.3542 &&& w & \approx 7.6458 \\ w & \approx 2.35 \text{ (2 d.p.)} &&& w & \approx 7.65 \text{ (2 d.p.)} \end{align}

(h)

\begin{align} \left({1 \over 2} - t\right)^2 & = 10 \\ {1 \over 2} - t & = \pm \sqrt{10} \end{align} \begin{align} {1 \over 2} - t & = \sqrt{10} && \text{ or } & {1 \over 2} - t & = - \sqrt{10} \\ -t & = \sqrt{10} - {1 \over 2} &&& -t & = -\sqrt{10} - {1 \over 2} \\ t & = - \sqrt{10} + {1 \over 2} &&& t & = \sqrt{10} + {1 \over 2} \\ t & \approx -2.6623 &&& t & \approx 3.6623 \\ t & \approx -2.66 \text{ (2 d.p.)} &&& t & \approx 3.66 \text{ (2 d.p.)} \end{align}

(a)

\begin{align} x^2 + 20x & = x^2 + 20x + \left(20 \over 2\right)^2 - \left(20 \over 2\right)^2 \\ & = x^2 + 20x + 10^2 - 10^2 \\ & = (x + 10)^2 - 10^2 \\ & = (x + 10)^2 - 100 \end{align}

(b)

\begin{align} x^2 - 15x & = x^2 - 15x + \left(15 \over 2\right)^2 - \left(15 \over 2\right)^2 \\ & = \left(x - {15 \over 2} \right)^2 - \left(15 \over 2\right)^2 \\ & = \left(x - {15 \over 2}\right)^2 - {225 \over 4} \end{align}

(c)

\begin{align} x^2 + {1 \over 2}x & = x^2 + {1 \over 2}x + \left({1 \over 2} \over 2\right)^2 - \left({1 \over 2} \over 2\right)^2 \\ & = x^2 + {1 \over 2}x + \left(1 \over 4\right)^2 - \left(1 \over 4\right)^2 \\ & = \left(x + {1 \over 4}\right)^2 - {1 \over 16} \end{align}

(d)

\begin{align} x^2 - {2 \over 9}x & = x^2 - {2 \over 9}x + \left({2 \over 9} \over 2\right)^2 - \left({2 \over 9} \over 2\right)^2 \\ & = x^2 - {2 \over 9}x + \left(1 \over 9\right)^2 - \left(1 \over 9\right)^2 \\ & = \left(x - {1 \over 9}\right)^2 - {1 \over 81} \end{align}

(e)

\begin{align} x^2 + 0.2x & = x^2 + 0.2x + \left(0.2 \over 2\right)^2 - \left(0.2 \over 2\right)^2 \\ & = x^2 + 0.2x + 0.1^2 - 0.1^2 \\ & = (x + 0.1)^2 - 0.1^2 \\ & = (x + 0.1)^2 - 0.01 \end{align}

(f)

\begin{align} x^2 - 1.4x & = x^2 - 1.4x + \left(1.4 \over 2\right)^2 - \left(1.4 \over 2\right)^2 \\ & = x^2 - 1.4x + 0.7^2 - 0.7^2 \\ & = (x - 0.7)^2 - 0.7^2 \\ & = (x - 0.7)^2 - 0.49 \end{align}

(g)

\begin{align} -x^2 - 10x & = -(x^2 + 10x) \\ & = - \left[ x^2 + 10x + \left(10 \over 2\right)^2 - \left(10 \over 2\right)^2 \right] \\ & = - ( x^2 + 10x + 5^2 - 5^2 ) \\ & = - [ (x + 5)^2 - 5^2 ] \\ & = - [ (x + 5)^2 - 25 ] \\ & = - (x + 5)^2 + 25 \end{align}

(h)

\begin{align} -x^2 + 11x & = -(x^2 - 11x) \\ & = - \left[ x^2 - 11x + \left(11 \over 2\right)^2 - \left(11 \over 2\right)^2 \right] \\ & = - \left[ \left(x - {11 \over 2}\right)^2 - \left(11 \over 2\right)^2 \right] \\ & = - \left[ \left(x - {11 \over 2}\right)^2 - {121 \over 4} \right] \\ & = - \left(x - {11 \over 2}\right)^2 + {121 \over 4} \end{align}

(a)

\begin{align} x^2 - 6x + 1 & = x^2 - 6x + \left(6 \over 2\right)^2 - \left(6 \over 2\right)^2 + 1 \\ & = x^2 - 6x + 3^2 - 3^2 + 1 \\ & = (x - 3)^2 - 3^2 + 1 \\ & = (x - 3)^2 - 8 \end{align}

(b)

\begin{align} x^2 + 3x - 2 & = x^2 + 3x + \left(3 \over 2\right)^2 - \left(3 \over 2\right)^2 - 2 \\ & = \left(x + {3 \over 2}\right)^2 - \left( 3 \over 2 \right)^2 - 2 \\ & = \left(x + {3 \over 2}\right)^2 - {17 \over 4} \end{align}

(c)

\begin{align} x^2 + 9x - 1.8 & = x^2 + 9x + \left(9 \over 2\right)^2 - \left(9 \over 2\right)^2 - 1.8 \\ & = x^2 + 9x + 4.5^2 - 4.5^2 - 1.8 \\ & = (x + 4.5)^2 - 4.5^2 - 1.8 \\ & = (x + 4.5)^2 - 22.05 \end{align}

(d)

\begin{align} x^2 - {2 \over 7}x + 7 & = x^2 - {2 \over 7}x + \left( {2 \over 7} \over 2\right)^2 - \left( {2 \over 7} \over 2\right)^2 + 7 \\ & = x^2 - {2 \over 7}x + \left(1 \over 7\right)^2 - \left(1 \over 7\right)^2 + 7 \\ & = \left(x - {1 \over 7}\right)^2 - \left(1 \over 7\right)^2 + 7 \\ & = \left(x - {1 \over 7}\right)^2 + {342 \over 49} \end{align}

(e)

\begin{align} -x^2 + 10x - 2 & = - (x^2 - 10x) - 2 \\ & = - \left[ x^2 - 10x + \left(10 \over 2\right)^2 - \left(10 \over 2\right)^2 \right] - 2 \\ & = - ( x^2 - 10x + 5^2 - 5^2 ) - 2 \\ & = - [ (x - 5)^2 - 5^2 ] - 2 \\ & = - [ (x - 5)^2 - 25 ] - 2 \\ & = - (x - 5)^2 + 25 - 2 \\ & = - (x - 5)^2 + 23 \end{align}

(f)

\begin{align} -x^2 + 13x - {13 \over 2} & = - (x^2 - 13x) - {13 \over 2} \\ & = - \left[ x^2 - 13x + \left(13 \over 2\right)^2 - \left(13 \over 2\right)^2 \right] - {13 \over 2} \\ & = - \left[ \left(x - {13 \over 2}\right)^2 - \left(13 \over 2\right)^2 \right] - {13 \over 2} \\ & = - \left[ \left(x - {13 \over 2}\right)^2 - {169 \over 4} \right] - {13 \over 2} \\ & = - \left(x - {13 \over 2}\right)^2 + {169 \over 4} - {13 \over 2} \\ & = - \left(x - {13 \over 2}\right)^2 + {143 \over 4} \end{align}

(g)

\begin{align} -x^2 - 9x - 20.25 & = -(x^2 + 9x) - 20.25 \\ & = - \left[ x^2 + 9x + \left(9 \over 2\right)^2 - \left(9 \over 2\right)^2 \right] - 20.25 \\ & = - (x^2 + 9x + 4.5^2 - 4.5^2) - 20.25 \\ & = - [ (x + 4.5)^2 - 4.5^2 ] - 20.25 \\ & = - [ (x + 4.5)^2 - 20.25 ] - 20.25 \\ & = - (x + 4.5)^2 + 20.25 - 20.25 \\ & = - (x + 4.5)^2 \end{align}

(h)

\begin{align} -x^2 - {3 \over 4}x + 3 & = - \left(x^2 + {3 \over 4}x \right) + 3 \\ & = - \left[ x^2 + {3 \over 4}x + \left({3 \over 4} \over 2\right)^2 - \left({3 \over 4} \over 2\right)^2 \right] + 3 \\ & = - \left[ x^2 + {3 \over 4}x + \left(3 \over 8\right)^2 - \left(3 \over 8\right)^2 \right] + 3 \\ & = - \left[ \left(x + {3 \over 8}\right)^2 - \left(3 \over 8\right)^2 \right] + 3 \\ & = - \left[ \left(x + {3 \over 8}\right)^2 - {9 \over 64} \right] + 3 \\ & = - \left(x + {3 \over 8}\right)^2 + {9 \over 64} + 3 \\ & = - \left(x + {3 \over 8}\right)^2 + {201 \over 64} \end{align}

(a)

\begin{align} x^2 + 2x - 5 & = 0 \\ x^2 + 2x + \left(2 \over 2\right)^2 - \left(2 \over 2\right)^2 - 5 & = 0 \\ x^2 + 2x + 1^2 - 1^2 - 5 & = 0 \\ (x + 1)^2 - 1^2 - 5 & = 0 \\ (x + 1)^2 - 6 & = 0 \\ (x + 1)^2 & = 6 \\ x + 1 & = \pm \sqrt{6} \end{align} \begin{align} x + 1 & = \sqrt{6} && \text{ or } & x + 1 & = - \sqrt{6} \\ x & = \sqrt{6} - 1 &&& x & = -\sqrt{6} - 1 \\ x & \approx 1.4495 &&& x & \approx -3.4495 \\ x & \approx 1.45 \text{ (2 d.p.)} &&& x & \approx -3.45 \text{ (2 d.p.)} \end{align}

(b)

\begin{align} y^2 - 12y + 9 & = 0 \\ y^2 - 12y + \left(12 \over 2\right)^2 - \left(12 \over 2\right)^2 + 9 & = 0 \\ y^2 - 12y + 6^2 - 6^2 + 9 & = 0 \\ (y - 6)^2 - 6^2 + 9 & = 0 \\ (y - 6)^2 - 27 & = 0 \\ (y - 6)^2 & = 27 \\ y - 6 & = \pm \sqrt{27} \end{align} \begin{align} y - 6 & = \sqrt{27} && \text{ or } & y - 6 & = -\sqrt{27} \\ y & = \sqrt{27} + 6 &&& y & = - \sqrt{27} + 6 \\ y & \approx 11.196 &&& y & \approx 0.80385 \\ y & \approx 11.20 \text{ (2 d.p.)} &&& y & \approx 0.80 \text{ (2 d.p.)} \end{align}

(c)

\begin{align} z^2 - 5z - 5 & = 0 \\ z^2 - 5z + \left(5 \over 2\right)^2 - \left(5 \over 2\right)^2 - 5 & = 0 \\ z^2 - 5z + 2.5^2 - 2.5^2 - 5 & = 0 \\ (z - 2.5)^2 - 2.5^2 - 5 & = 0 \\ (z - 2.5)^2 - 11.25 & = 0 \\ (z - 2.5)^2 & = 11.25 \\ z - 2.5 & = \pm \sqrt{11.25} \end{align} \begin{align} z - 2.5 & = \sqrt{11.25} && \text{ or } & z - 2.5 & = - \sqrt{11.25} \\ z & = \sqrt{11.25} + 2.5 &&& z & = - \sqrt{11.25} + 2.5 \\ z & \approx 5.8541 &&& z & \approx -0.85410 \\ z & \approx 5.85 \text{ (2 d.p.)} &&& z & \approx -0.85 \text{ (2 d.p.)} \end{align}

(d)

\begin{align} p^2 + {1 \over 4}p - 3 & = 0 \\ p^2 + {1 \over 4}p + \left({1 \over 4} \over 2\right)^2 - \left({1 \over 4} \over 2\right)^2 - 3 & = 0 \\ p^2 + {1 \over 4}p + \left(1 \over 8\right)^2 - \left(1 \over 8\right)^2 - 3 & = 0 \\ \left(p + {1 \over 8}\right)^2 - \left(1 \over 8\right)^2 - 3 & = 0 \\ \left(p + {1 \over 8}\right)^2 - {193 \over 64} & = 0 \\ \left(p + {1 \over 8}\right)^2 & = {193 \over 64} \\ p + {1 \over 8} & = \pm \sqrt{193 \over 64} \end{align} \begin{align} p + {1 \over 8} & = \sqrt{193 \over 64} && \text{ or } & p + {1 \over 8} & = - \sqrt{193 \over 64} \\ p & = \sqrt{193 \over 64} - {1 \over 8} &&& p & = - \sqrt{193 \over 64} - {1 \over 8} \\ p & \approx 1.6116 &&& p & \approx -1.8616 \\ p & \approx 1.61 \text{ (2 d.p.)} &&& p & \approx -1.86 \text{ (2 d.p.)} \end{align}

(e)

\begin{align} q^2 - {6 \over 7}q + {2 \over 49} & = 0 \\ q^2 - {6 \over 7}q + \left({6 \over 7} \over 2\right)^2 - \left({6 \over 7} \over 2\right)^2 + {2 \over 49} & = 0 \\ q^2 - {6 \over 7}q + \left(3 \over 7\right)^2 - \left(3 \over 7\right)^2 + {2 \over 49} & = 0 \\ \left(q - {3 \over 7}\right)^2 - \left(3 \over 7\right)^2 + {2 \over 49} & = 0 \\ \left(q - {3 \over 7}\right)^2 - {1 \over 7} & = 0 \\ \left(q - {3 \over 7}\right)^2 & = {1 \over 7} \\ q - {3 \over 7} & = \pm \sqrt{1 \over 7} \end{align} \begin{align} q - {3 \over 7} & = \sqrt{1 \over 7} && \text{ or } & q - {3 \over 7} & = - \sqrt{1 \over 7} \\ q & = \sqrt{1 \over 7} + {3 \over 7} &&& q & = - \sqrt{1 \over 7} + {3 \over 7} \\ q & \approx 0.80654 &&& q & \approx 0.050607 \\ q & \approx 0.81 \text{ (2 d.p.)} &&& q & \approx 0.05 \text{ (2 d.p.)} \end{align}

(f)

\begin{align} r^2 + 0.6r - 1 & = 0 \\ r^2 + 0.6r + \left(0.6 \over 2 \right)^2 - \left(0.6 \over 2\right)^2 - 1 & = 0 \\ r^2 + 0.6r + 0.3^2 - 0.3^2 - 1 & = 0 \\ (r + 0.3)^2 - 0.3^2 - 1 & = 0 \\ (r + 0.3)^2 - 1.09 & = 0 \\ (r + 0.3)^2 & = 1.09 \\ r + 0.3 & = \pm \sqrt{1.09} \end{align} \begin{align} r + 0.3 & = \sqrt{1.09} && \text{ or } & r + 0.3 & = -\sqrt{1.09} \\ r & = \sqrt{1.09} - 0.3 &&& r & = - \sqrt{1.09} - 0.3 \\ r & \approx 0.74403 &&& r & \approx -1.3440 \\ r & \approx 0.74 \text{ (2 d.p.)} &&& r & \approx -1.34 \text{ (2 d.p.)} \end{align}

(i)

\begin{align} x^2 + 17x - 30 & = x^2 + 17x + \left(17 \over 2\right)^2 - \left(17 \over 2\right)^2 - 30 \\ & = \left(x + {17 \over 2}\right)^2 - \left(17 \over 2\right)^2 - 30 \\ & = \left(x + {17 \over 2}\right)^2 - {409 \over 4} \end{align}

(ii)

\begin{align} x^2 + 17x - 30 & = 0 \\ \\ \text{From (i), } x^2 + 17x - 30 & = \left(x + {17 \over 2}\right)^2 - {409 \over 4} \\ \\ \\ \left(x + {17 \over 2}\right)^2 - {409 \over 4} & = 0 \\ \left(x + {17 \over 2}\right)^2 & = {409 \over 4} \\ x + {17 \over 2} & = \pm \sqrt{409 \over 4} \end{align} \begin{align} x + {17 \over 2} & = \sqrt{409 \over 4} && \text{ or } & x + {17 \over 2} & = - \sqrt{409 \over 4} \\ x & = \sqrt{409 \over 4} - {17 \over 2} &&& x & = - \sqrt{409 \over 4} - {17 \over 2} \\ x & \approx 1.6119 &&& x & \approx -18.612 \\ x & \approx 1.6 \text{ (1 d.p.)} &&& x & \approx -18.6 \text{ (1 d.p.)} \end{align}

(a)

\begin{align} a(a + 4) & = 3a + 1 \\ a^2 + 4a & = 3a + 1 \\ a^2 + 4a - 3a - 1 & = 0 \\ a^2 + a - 1 & = 0 \\ a^2 + a + \left(1 \over 2\right)^2 - \left(1 \over 2\right)^2 - 1 & = 0\\ a^2 + a + 0.5^2 - 0.5^2 - 1 & = 0 \\ (a + 0.5)^2 - 0.5^2 - 1 & = 0 \\ (a + 0.5)^2 - 1.25 & = 0 \\ (a + 0.5)^2 & = 1.25 \\ a + 0.5 & = \pm \sqrt{1.25} \end{align} \begin{align} a + 0.5 & = \sqrt{1.25} && \text{ or } & a + 0.5 & = -\sqrt{1.25} \\ a & = \sqrt{1.25} - 0.5 &&& a & = -\sqrt{1.25} - 0.5 \\ a & \approx 0.61803 &&& a & \approx -1.6180 \\ a & \approx 0.618 &&& a & \approx -1.62 \end{align}

(b)

\begin{align} (b + 1)^2 & = 7b \\ (b)^2 + 2(b)(1) + (1)^2 & = 7b \phantom{000000} [(a + b)^2 = a^2 + 2ab + b^2] \\ b^2 + 2b + 1 & = 7b \\ b^2 + 2b - 7b + 1 & = 0 \\ b^2 - 5b + 1 & = 0 \\ b^2 - 5b + \left(5 \over 2\right)^2 - \left(5 \over 2\right)^2 + 1 & = 0 \\ b^2 - 5b + 2.5^2 - 2.5^2 + 1 & = 0 \\ (b - 2.5)^2 - 2.5^2 + 1 & = 0 \\ (b - 2.5)^2 - 5.25 & = 0 \\ (b - 2.5)^2 & = 5.25 \\ b - 2.5 & = \pm \sqrt{5.25} \end{align} \begin{align} b - 2.5 & = \sqrt{5.25} && \text{ or } & b - 2.5 & = - \sqrt{5.25} \\ b & = \sqrt{5.25} + 2.5 &&& b & = - \sqrt{5.25} + 2.5 \\ b & \approx 4.7913 &&& b & \approx 0.20871 \\ b & \approx 4.79 &&& b & \approx 0.209 \end{align}

(c)

\begin{align} (c - 2)(c + 5) & = c \\ c^2 + 5c - 2c - 10 & = c \\ c^2 + 3c - 10 & = c \\ c^2 + 3c - c - 10 & = 0 \\ c^2 + 2c - 10 & = 0 \\ c^2 + 2c + \left(2 \over 2\right)^2 - \left(2 \over 2\right)^2 - 10 & = 0 \\ c^2 + 2c + 1^2 - 1^2 - 10 & = 0 \\ (c + 1)^2 - 1^2 - 10 & = 0 \\ (c + 1)^2 - 11 & = 0 \\ (c + 1)^2 & = 11 \\ c + 1 & = \pm \sqrt{11} \end{align} \begin{align} c + 1 & = \sqrt{11} && \text{ or } & c + 1 & = - \sqrt{11} \\ c & = \sqrt{11} - 1 &&& c & = - \sqrt{11} - 1 \\ c & \approx 2.3166 &&& c & \approx -4.3166 \\ c & \approx 2.32 &&& c & \approx -4.32 \end{align}

(d)

\begin{align} d(d - 4) & = 2(d + 7) \\ d^2 - 4d & = 2d + 14 \\ d^2 - 4d - 2d - 14 & = 0 \\ d^2 - 6d - 14 & = 0 \\ d^2 - 6d + \left(6 \over 2\right)^2 - \left(6 \over 2\right)^2 - 14 & = 0 \\ d^2 - 6d + 3^2 - 3^2 - 14 & = 0 \\ (d - 3)^2 - 3^2 - 14 & = 0 \\ (d - 3)^2 - 23 & = 0 \\ (d - 3)^2 & = 23 \\ d - 3 & = \pm \sqrt{23} \end{align} \begin{align} d - 3 & = \sqrt{23} && \text{ or } & d - 3 & = - \sqrt{23} \\ d & = \sqrt{23} + 3 &&& d & = - \sqrt{23} + 3 \\ d & \approx 7.7958 &&& d & \approx -1.7958 \\ d & \approx 7.80 &&& d & \approx -1.80 \end{align}

\begin{align} y^2 - ay - 6 & = 0 \\ y^2 - ay + \left(a \over 2\right)^2 - \left(a \over 2\right)^2 - 6 & = 0 \\ \left( y - {a \over 2} \right)^2 - \left(a \over 2\right)^2 - 6 & = 0 \\ \left( y - {a \over 2} \right)^2 - {a^2 \over 4} - 6 & = 0 \\ \left( y - {a \over 2} \right)^2 & = {a^2 \over 4} + 6 \\ \left( y - {a \over 2} \right)^2 & = {a^2 \over 4} + {6 \over 1} \\ \left( y - {a \over 2} \right)^2 & = {a^2 \over 4} + {24 \over 4} \\ \left( y - {a \over 2} \right)^2 & = {a^2 + 24 \over 4} \\ y - {a \over 2} & = \pm \sqrt{ a^2 + 24 \over 4 } \\ y - {a \over 2} & = { \pm \sqrt{a^2 + 24} \over \sqrt{4} } \\ y - {a \over 2} & = { \pm \sqrt{a^2 + 24} \over 2 } \\ y & = { \pm \sqrt{a^2 + 24} \over 2} + {a \over 2} \\ y & = { \pm \sqrt{a^2 + 24} + a \over 2 } \\ y & = {a \pm \sqrt{a^2 + 24} \over 2} \end{align}

(a)

\begin{align} x^2 & + 4x + 1 = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 4 \pm \sqrt{(4)^2 - 4(1)(1)} \over 2(1) } \\ & = { - 4 \pm \sqrt{12} \over 2} \\ & \approx -0.26795 \text{ or } -3.7321 \\ & \approx -0.268 \text{ or } -3.73 \end{align}

(b)

\begin{align} 3x^2 & + 6x - 1 = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 6 \pm \sqrt{(6)^2 - 4(3)(-1)} \over 2(3) } \\ & = { - 6 \pm \sqrt{48} \over 6} \\ & \approx 0.15470 \text{ or } -2.1547 \\ & \approx 0.155 \text{ or } -2.15 \end{align}

(c)

\begin{align} 3x^2 & - 5x - 17 = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-5) \pm \sqrt{(-5)^2 - 4(3)(-17)} \over 2(3) } \\ & = { 5 \pm \sqrt{229} \over 6} \\ & \approx 3.3555 \text{ or } -1.6888 \\ & \approx 3.36 \text{ or } -1.69 \end{align}

(d)

\begin{align} -3x^2 & - 7x + 9 = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-7) \pm \sqrt{(-7)^2 - 4(-3)(9)} \over 2(-3) } \\ & = { 7 \pm \sqrt{157} \over -6} \\ & \approx -3.25499 \text{ or } 0.92166 \\ & \approx -3.25 \text{ or } 0.922 \end{align}

(e)

\begin{align} 2 + 2x^2 - 7x & = 0 \\ 2x^2 - 7x + 2 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-7) \pm \sqrt{(-7)^2 - 4(2)(2)} \over 2(2) } \\ & = { 7 \pm \sqrt{33} \over 4} \\ & \approx 3.1861 \text{ or } 0.31386 \\ & \approx 3.19 \text{ or } 0.314 \end{align}

(f)

\begin{align} 10x - 5x^2 - 2 & = 0 \\ - 5x^2 + 10x - 2 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 10 \pm \sqrt{(10)^2 - 4(-5)(-2)} \over 2(-5) } \\ & = { -10 \pm \sqrt{60} \over -10} \\ & \approx 0.22540 \text{ or } 1.7746 \\ & \approx 0.225 \text{ or } 1.77 \end{align}

(a)

\begin{align} x^2 + 5x & = 21 \\ x^2 + 5x - 21 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 5 \pm \sqrt{(5)^2 - 4(1)(-21)} \over 2(1) } \\ & = { -5 \pm \sqrt{109} \over 2} \\ & \approx 2.7202 \text{ or } -7.7202 \\ & \approx 2.72 \text{ or } -7.72 \end{align}

(b)

\begin{align} 10x^2 - 12x & = 15 \\ 10x^2 - 12x - 15 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-12) \pm \sqrt{(-12)^2 - 4(10)(-15)} \over 2(10) } \\ & = { 12 \pm \sqrt{744} \over 20} \\ & \approx 1.9638 \text{ or } -0.76382 \\ & \approx 1.96 \text{ or } -0.764 \end{align}

(c)

\begin{align} 8x^2 & = 3x + 6 \\ 8x^2 - 3x - 6 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-3) \pm \sqrt{(-3)^2 - 4(8)(-6)} \over 2(8) } \\ & = { 3 \pm \sqrt{201} \over 16} \\ & \approx 1.0736 \text{ or } -0.69859 \\ & \approx 1.07 \text{ or } -0.699 \end{align}

(d)

\begin{align} 4x^2 + 1 & = - 4x \\ 4x^2 + 4x + 1 & = 0 \\ (2x + 1)(2x + 1) & = 0 \\ (2x + 1)^2 & = 0 \\ 2x + 1 & = \sqrt{0} \\ 2x + 1 & = 0 \\ 2x & = -1 \\ x & = {-1 \over 2} \\ x & = -0.5 \end{align}

(e)

\begin{align} 9 - 5x^2 & = -3x \\ 0 & = 5x^2 - 3x - 9 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-3) \pm \sqrt{(-3)^2 - 4(5)(-9)} \over 2(5) } \\ & = { 3 \pm \sqrt{189} \over 10} \\ & \approx 1.6748 \text{ or } -1.0748 \\ & \approx 1.67 \text{ or } -1.07 \end{align}

(f)

\begin{align} 16x - 61 & = x^2 \\ 0 & = x^2 - 16x + 61 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-16) \pm \sqrt{(-16)^2 - 4(1)(61)} \over 2(1) } \\ & = { 16 \pm \sqrt{12} \over 2} \\ & \approx 9.7321 \text{ or } 6.2679 \\ & \approx 9.73 \text{ or } 6.27 \end{align}

(a)

\begin{align} {8 \over x} & = 2x + 1 \\ {8 \over x} & = {2x + 1 \over 1} \\ 8 & = x(2x + 1) \\ 8 & = 2x^2 + x \\ 0 & = 2x^2 + x - 8 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 1 \pm \sqrt{(1)^2 - 4(2)(-8)} \over 2(2) } \\ & = { -1 \pm \sqrt{65} \over 4} \\ & \approx 1.7656 \text{ or } -2.2656 \\ & \approx 1.77 \text{ or } -2.27 \end{align}

(b)

\begin{align} x + {7 \over x} & = 9 \\ {7 \over x} & = 9 - x \\ {7 \over x} & = {9 - x \over 1} \\ 7 & = x(9 - x) \\ 7 & = 9x - x^2 \\ 0 & = -x^2 + 9x - 7 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 9 \pm \sqrt{(9)^2 - 4(-1)(-7)} \over 2(-1) } \\ & = { -9 \pm \sqrt{53} \over -2} \\ & \approx 0.85995 \text{ or } 8.1401 \\ & \approx 0.860 \text{ or } 8.14 \end{align}

(c)

\begin{align} {x + 1 \over 5 - x} & = x \\ {x + 1 \over 5 - x} & = {x \over 1} \\ x + 1 & = x(5 - x) \\ x + 1 & = 5x - x^2 \\ 0 & = - x^2 + 5x - x - 1 \\ 0 & = - x^2 + 4x - 1 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 4 \pm \sqrt{(4)^2 - 4(-1)(-1)} \over 2(-1) } \\ & = { -4 \pm \sqrt{12} \over -2} \\ & \approx 0.26795 \text{ or } 3.7321 \\ & \approx 0.268 \text{ or } 3.73 \end{align}

(d)

\begin{align} {3x - 1} & = -{9 \over 3x + 5} \\ {3x - 1 \over 1} & = {-9 \over 3x + 5} \\ (3x - 1)(3x + 5) & = - 9 \\ 9x^2 + 15x - 3x - 5 & = - 9 \\ 9x^2 + 12x - 5 + 9 & = 0 \\ 9x^2 + 12x + 4 & = 0 \\ (3x + 2)(3x + 2) & = 0 \\ (3x + 2)^2 & = 0 \\ 3x + 2 & = \sqrt{0} \\ 3x + 2 & = 0 \\ 3x & = -2 \\ x & = -{2 \over 3} \end{align}

(e)

\begin{align} 2x + 1 & = {x + 1 \over x - 5} \\ {2x + 1 \over 1} & = {x + 1 \over x - 5} \\ (2x + 1)(x - 5) & = x + 1 \\ 2x^2 - 10x + x - 5 & = x + 1 \\ 2x^2 - 9x - 5 & = x + 1 \\ 2x^2 - 9x - x - 5 - 1 & = 0 \\ 2x^2 - 10x - 6 & = 0 \\ x^2 - 5x - 3 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-5) \pm \sqrt{(-5)^2 - 4(1)(-3)} \over 2(1) } \\ & = { 5 \pm \sqrt{37} \over 2} \\ & \approx 5.5414 \text{ or } -0.54138 \\ & \approx 5.54 \text{ or } -0.541 \end{align}

(f)

\begin{align} {5x \over x + 4} & = 4x + 1 \\ {5x \over x + 4} & = {4x + 1 \over 1} \\ 5x & = (4x + 1)(x + 4) \\ 5x & = 4x^2 + 16x + x + 4 \\ 5x & = 4x^2 + 17x + 4 \\ 0 & = 4x^2 + 17x - 5x + 4 \\ 0 & = 4x^2 + 12x + 4 \\ 0 & = x^2 + 3x + 1 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 3 \pm \sqrt{(3)^2 - 4(1)(1)} \over 2(1) } \\ & = { -3 \pm \sqrt{5} \over 2} \\ & \approx -0.38197 \text{ or } -2.6180 \\ & \approx -0.382 \text{ or } -2.62 \end{align}

(a)

\begin{align} x(x + 1) & = 1 \\ x^2 + x & = 1 \\ x^2 + x - 1 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 1 \pm \sqrt{(1)^2 - 4(1)(-1)} \over 2(1) } \\ & = { -1 \pm \sqrt{5} \over 2} \\ & \approx 0.61803 \text{ or } -1.6180 \\ & \approx 0.618 \text{ or } -1.62 \end{align}

(b)

\begin{align} 3(x + 1)(x - 1) & = 7x \\ 3(x^2 - 1^2) & = 7x \phantom{000000} [(a + b)(a - b) = a^2 - b^2] \\ 3(x^2 - 1) & = 7x \\ 3x^2 - 3 & = 7x \\ 3x^2 - 7x - 3 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-7) \pm \sqrt{(-7)^2 - 4(3)(-3)} \over 2(3) } \\ & = { 7 \pm \sqrt{85} \over 6} \\ & \approx 2.7033 \text{ or } -0.36992 \\ & \approx 2.70 \text{ or } -0.370 \end{align}

(c)

\begin{align} (x - 1)^2 - 2x & = 0 \\ (x)^2 - 2(x)(1) + (1)^2 - 2x & = 0 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ x^2 - 2x + 1 - 2x & = 0 \\ x^2 - 4x + 1 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-4) \pm \sqrt{(-4)^2 - 4(1)(1)} \over 2(1) } \\ & = { 4 \pm \sqrt{12} \over 2} \\ & \approx 3.7321 \text{ or } 0.26795 \\ & \approx 3.73 \text{ or } 0.268 \end{align}

(d)

\begin{align} x(x - 5) & = 7 - 2x \\ x^2 - 5x & = 7 - 2x \\ x^2 - 5x + 2x - 7 & = 0 \\ x^2 - 3x - 7 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-3) \pm \sqrt{(-3)^2 - 4(1)(-7)} \over 2(1) } \\ & = { 3 \pm \sqrt{37} \over 2} \\ & \approx 4.5414 \text{ or } -1.5414 \\ & \approx 4.54 \text{ or } -1.54 \end{align}

(e)

\begin{align} (5x - 9)(x - 1) - x(x - 2) & = 0 \\ 5x^2 - 5x - 9x + 9 - x^2 + 2x & = 0 \\ 5x^2 - x^2 - 5x - 9x + 2x + 9 & = 0 \\ 4x^2 - 12x + 9 & = 0 \\ (2x - 3)(2x - 3) & = 0 \\ (2x - 3)^2 & = 0 \\ 2x - 3 & = \sqrt{0} \\ 2x - 3 & = 0 \\ 2x & = 3 \\ x & = {3 \over 2} \\ x & = 1.5 \end{align}

(f)

\begin{align} (4x - 3)^2 + (4x + 3)^2 & = 25 \\ \underbrace{(4x)^2 - 2(4x)(3) + 3^2}_{(a - b)^2 = a^2 - 2ab + b^2} + \underbrace{(4x)^2 + 2(4x)(3) + (3)^2}_{(a + b)^2 = a^2 + 2ab + b^2} & = 25 \\ \\ 16x^2 - 24x + 9 + 16x^2 + 24x + 9 & = 25 \\ 32x^2 + 18 & = 25 \\ 32x^2 & = 25 - 18 \\ 32x^2 & = 7 \\ x^2 & = {7 \over 32} \\ x & = \pm \sqrt{7 \over 32} \\ x & \approx \pm 0.46771 \\ x & \approx \pm 0.468 \end{align}

(a)

\begin{align} {x - 1 \over x + 1} & = {8x \over 1 - x} \\ (x - 1)(1 - x) & = 8x(x + 1) \\ x - x^2 - 1 + x & = 8x^2 + 8x \\ 2x - x^2 - 1 & = 8x^2 + 8x \\ 0 & = 8x^2 + x^2 + 8x - 2x + 1 \\ 0 & = 9x^2 + 6x + 1 \\ 0 & = (3x + 1)(3x + 1) \\ 0 & = (3x + 1)^2 \\ \sqrt{0} & = 3x + 1 \\ 0 & = 3x + 1 \\ -1 & = 3x \\ -{1 \over 3} & = x \end{align}

(b)

\begin{align} { (x - 2)(x - 3) \over (x - 1)(x + 2)} & = {2 \over 3} \\ { x^2 - 3x - 2x + 6 \over x^2 + 2x - x - 2 } & = {2 \over 3} \\ { x^2 - 5x + 6 \over x^2 + x - 2} & = {2 \over 3} \\ 3(x^2 - 5x + 6) & = 2(x^2 + x - 2) \\ 3x^2 - 15x + 18 & = 2x^2 + 2x - 4 \\ 3x^2 - 2x^2 - 15x - 2x + 18 + 4 & = 0 \\ x^2 - 17x + 22 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-17) \pm \sqrt{(-17)^2 - 4(1)(22)} \over 2(1) } \\ & = { 17 \pm \sqrt{201} \over 2} \\ & \approx 15.589 \text{ or } 1.4113 \\ & \approx 15.6 \text{ or } 1.41 \end{align}

\begin{align} {x(x - 3) \over (x + 1)^2} & = {3 \over 5} \\ {x^2 - 3x \over x^2 + 2(x)(1) + (1)^2} & = {3 \over 5} \phantom{000000} [(a + b)^2 = a^2 + 2ab + b^2] \\ {x^2 - 3x \over x^2 + 2x + 1} & = {3 \over 5} \\ 5(x^2 - 3x) & = 3(x^2 + 2x + 1) \\ 5x^2 - 15x & = 3x^2 + 6x + 3 \\ 5x^2 - 3x^2 - 15x - 6x - 3 & = 0 \\ 2x^2 - 21x - 3 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-21) \pm \sqrt{(-21)^2 - 4(2)(-3)} \over 2(2) } \\ & = { 21 \pm \sqrt{465} \over 4} \\ & \approx 10.641 \text{ or } -0.14096 \\ & \approx 10.6 \text{ or } -0.141 \end{align}

(a)

\begin{align} {x \over 2} & = {4 \over x} - 1 \\ {x \over 2} & = {4 \over x} - {x \over x} \\ {x \over 2} & = {4 - x \over x} \\ x(x) & = 2(4 - x) \\ x^2 & = 8 - 2x \\ x^2 + 2x - 8 & = 0 \\ (x + 4)(x - 2) & = 0 \end{align} \begin{align} x + 4 & = 0 && \text{ or } & x - 2 & = 0 \\ x & = -4 &&& x & = 2 \end{align}

(b)

\begin{align} {2 \over x + 5} & = 1 - {x + 1 \over 5} \\ {2 \over x + 5} & = {5 \over 5} - {x + 1 \over 5} \\ {2 \over x + 5} & = {5 - (x + 1) \over 5} \\ {2 \over x + 5} & = {5 - x - 1 \over 5} \\ {2 \over x + 5} & = {4 - x \over 5} \\ 5(2) & = (x + 5)(4 - x) \\ 10 & = 4x - x^2 + 20 - 5x \\ 10 & = - x^2 - x + 20 \\ 0 & = - x^2 - x + 20 - 10 \\ 0 & = - x^2 - x + 10 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-1) \pm \sqrt{(-1)^2 - 4(-1)(10)} \over 2(-1) } \\ & = { 1 \pm \sqrt{41} \over -2} \\ & \approx -3.7016 \text{ or } 2.7016 \\ & \approx -3.70 \text{ or } 2.70 \end{align}

(c)

\begin{align} {x - 2 \over 5} + {1 \over 2x - 3} & = 2 \\ {(x - 2)(2x - 3) \over 5(2x - 3)} + {5 \over 5(2x - 3)} & = 2 \\ {(x - 2)(2x - 3) + 5 \over 5(2x - 3)} & = 2 \\ {2x^2 - 3x - 4x + 6 + 5 \over 10x - 15} & = 2 \\ {2x^2 - 7x + 11 \over 10x - 15} & = {2 \over 1} \\ 2x^2 - 7x + 11 & = 2(10x - 15) \\ 2x^2 - 7x + 11 & = 20x - 30 \\ 2x^2 - 7x - 20x + 11 + 30 & = 0 \\ 2x^2 - 27x + 41 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-27) \pm \sqrt{(-27)^2 - 4(2)(41)} \over 2(2) } \\ & = { 27 \pm \sqrt{401} \over 4} \\ & \approx 11.756 \text{ or } 1.7438 \\ & \approx 11.8 \text{ or } 1.74 \end{align}

(d)

\begin{align} {4 \over x} - {1 \over x - 1} & = 9 \\ {4(x - 1) \over x(x - 1)} - {x \over x(x - 1)} & = 9 \\ {4(x - 1) - x \over x(x - 1)} & = 9 \\ {4x - 4 - x \over x^2 - x} & = 9 \\ {3x - 4 \over x^2 - x} & = {9 \over 1} \\ 3x - 4 & = 9(x^2 - x) \\ 3x - 4 & = 9x^2 - 9x \\ 0 & = 9x^2 - 9x - 3x + 4 \\ 0 & = 9x^2 - 12x + 4 \\ 0 & = (3x - 2)(3x - 2) \\ 0 & = (3x - 2)^2 \\ \sqrt{0} & = 3x - 2 \\ 0 & = 3x - 2 \\ 2 & = 3x \\ {2 \over 3} & = x \end{align}

(e)

\begin{align} {1 \over x + 2} + {1 \over x - 2} & = {3 \over 11} \\ {x - 2 \over (x + 2)(x - 2)} + {x + 2 \over (x + 2)(x - 2)} & = {3 \over 11} \\ {x - 2 + x + 2 \over (x + 2)(x - 2)} & = {3 \over 11} \\ {2x \over x^2 - 2^2} & = {3 \over 11} \phantom{000000} [(a + b)(a - b) = a^2 - b^2] \\ {2x \over x^2 - 4} & = {3 \over 11} \\ 11(2x) & = 3(x^2 - 4) \\ 22x & = 3x^2 - 12 \\ 0 & = 3x^2 - 22x - 12 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-22) \pm \sqrt{(-22)^2 - 4(3)(-12)} \over 2(3) } \\ & = { 22 \pm \sqrt{628} \over 6} \\ & \approx 7.8433 \text{ or } -0.50999 \\ & \approx 7.84 \text{ or } -0.510 \end{align}

(f)

\begin{align} {7 \over x - 1} - {x + 1 \over x + 3} & = {1 \over 2} \\ {7(x + 3) \over (x - 1)(x + 3)} - {(x + 1)(x - 1) \over (x - 1)(x + 3)} & = {1 \over 2} \\ {7(x + 3) - (x + 1)(x - 1) \over (x - 1)(x + 3)} & = {1 \over 2} \\ {7x + 21 - (x^2 - 1^2) \over x^2 + 3x - x - 3 } & = {1 \over 2} \phantom{000000} [(a + b)(a - b) = a^2 - b^2] \\ {7x + 21 - x^2 + 1 \over x^2 + 2x - 3} & = {1 \over 2} \\ {-x^2 + 7x + 22 \over x^2 + 2x - 3} & = {1 \over 2} \\ 2(-x^2 + 7x + 22) & = x^2 + 2x - 3 \\ -2x^2 + 14x + 44 & = x^2 + 2x - 3 \\ 0 & = x^2 + 2x^2 + 2x - 14x - 3 - 44 \\ 0 & = 3x^2 - 12x - 47 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-12) \pm \sqrt{(-12)^2 - 4(3)(-47)} \over 2(3) } \\ & = { 12 \pm \sqrt{708} \over 6} \\ & \approx 6.4347 \text{ or } -2.4347 \\ & \approx 6.43 \text{ or } -2.43 \end{align}

(g)

\begin{align} {5 \over x - 2} & = 2 - {4 \over (x - 2)^2} \\ {5 \over x - 2} + {4 \over (x - 2)^2} & = 2 \\ {5(x - 2) \over (x - 2)^2} + {4 \over (x - 2)^2} & = 2 \\ {5(x - 2) + 4 \over (x - 2)^2} & = 2 \\ {5x - 10 + 4 \over (x)^2 - 2(x)(2) + (2)^2} & = 2 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ {5x - 6 \over x^2 - 4x + 4} & = {2 \over 1} \\ 5x - 6 & = 2(x^2 - 4x + 4) \\ 5x - 6 & = 2x^2 - 8x + 8 \\ 0 & = 2x^2 - 8x - 5x + 8 + 6 \\ 0 & = 2x^2 - 13x + 14 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-13) \pm \sqrt{(-13)^2 - 4(2)(14)} \over 2(2) } \\ & = { 13 \pm \sqrt{57} \over 4} \\ & \approx 5.1375 \text{ or } 1.3625 \\ & \approx 5.14 \text{ or } 1.363 \end{align}

(h)

\begin{align} {5 \over x - 1} + {x \over (x - 1)^2} & = 1 \\ {5(x - 1) \over (x - 1)^2} + {x \over (x - 1)^2} & = 1 \\ {5(x - 1) + x \over (x - 1)^2} & = 1 \\ {5x - 5 + x \over (x)^2 - 2(x)(1) + (1)^2} & = 1 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ {6x -5 \over x^2 - 2x + 1} & = {1 \over 1} \\ 6x - 5 & = x^2 - 2x + 1 \\ 0 & = x^2 - 2x - 6x + 1 + 5 \\ 0 & = x^2 - 8x + 6 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-8) \pm \sqrt{(-8)^2 - 4(1)(6)} \over 2(1) } \\ & = { 8 \pm \sqrt{40} \over 2} \\ & \approx 7.1623 \text{ or } 0.83772 \\ & \approx 7.16 \text{ or } 0.838 \end{align}

(a)

\begin{align} {1 \over x} + {2 \over x - 1} + {3 \over x + 1} & = 0 \\ {2 \over x - 1} + {3 \over x + 1} & = - {1 \over x} \\ {2(x + 1) \over (x + 1)(x - 1)} + {3(x - 1) \over (x + 1)(x -1 )} & = -{1 \over x} \\ {2(x + 1) + 3(x - 1) \over (x + 1)(x - 1)} & = -{1 \over x} \\ {2x + 2 + 3x - 3 \over x^2 - 1^2} & = -{1 \over x} \phantom{000000} [(a + b)(a - b) = a^2 - b^2] \\ {5x - 1 \over x^2 - 1} & = {-1 \over x} \\ x(5x - 1) & = (-1)(x^2 - 1) \\ 5x^2 - x & = - x^2 + 1 \\ 5x^2 + x^2 - x - 1 & = 0 \\ 6x^2 - x - 1 & = 0 \\ (2x - 1)(3x + 1) & = 0 \end{align} \begin{align} 2x - 1 & = 0 && \text{ or } & 3x + 1 & =0 \\ 2x & = 1 &&& 3x & = -1 \\ x & = {1 \over 2} &&& x & = -{1 \over 3} \end{align}

(b)

\begin{align} {1 \over x^2 - 9} - {2 \over 3 - x} & = 1 \\ {1 \over (x + 3)(x - 3)} - {2 \over 3 - x} & = 1 \phantom{000000} [a^2 - b^2 = (a + b)(a - b)] \\ {1 \over (x + 3)(x - 3)} + {2 \over x - 3} & = 1 \\ {1 \over (x + 3)(x - 3)} + {2(x + 3) \over (x + 3)(x - 3)} & = 1 \\ {1 + 2(x + 3) \over (x + 3)(x - 3)} & = 1 \\ {1 + 2x + 6 \over x^2 - 9} & = 1 \\ {2x + 7 \over x^2 - 9} & = {1 \over 1} \\ 2x + 7 & = x^2 - 9 \\ 0 & = x^2 - 2x - 9 - 7 \\ 0 & = x^2 - 2x - 16 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-2) \pm \sqrt{(-2)^2 - 4(1)(-16)} \over 2(1) } \\ & = { 2 \pm \sqrt{68} \over 2} \\ & \approx 5.1231 \text{ or } -3.1231 \\ & \approx 5.12 \text{ or } -3.12 \end{align}

(c)

\begin{align} {3 \over x - 3} + {x + 1 \over x^2 - 5x + 6} & = 1 \\ {3 \over x - 3} + {x + 1 \over (x - 3)(x - 2)} & = 1 \\ {3(x - 2) \over (x - 3)(x - 2)} + {x + 1 \over (x - 3)(x - 2)} & = 1 \\ {3(x - 2) + x + 1 \over (x - 3)(x - 2)} & = 1 \\ {3x - 6 + x + 1 \over x^2 - 5x + 6} & = 1 \\ {4x - 5 \over x^2 - 5x + 6} & = {1 \over 1} \\ 4x - 5 & = x^2 - 5x + 6 \\ 0 & = x^2 - 5x - 4x + 6 + 5 \\ 0 & = x^2 - 9x + 11 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-9) \pm \sqrt{(-9)^2 - 4(1)(11)} \over 2(1) } \\ & = { 9 \pm \sqrt{37} \over 2} \\ & \approx 7.5414 \text{ or } 1.4586 \\ & \approx 7.54 \text{ or } 1.46 \end{align}

(d)

\begin{align} {4 \over x - 1} + 2 & = {x + 2 \over 2x^2 + 3x - 5} \\ {4 \over x - 1} + 2 & = {x + 2 \over (x - 1)(2x + 5)} \\ 2 & = {x + 2 \over (x - 1)(2x + 5)} - {4 \over x - 1} \\ 2 & = {x + 2 \over (x - 1)(2x + 5)} - {4(2x + 5) \over (x - 1)(2x + 5)} \\ 2 & = {x + 2 - 4(2x + 5) \over (x - 1)(2x + 5)} \\ 2 & = {x + 2 - 8x - 20 \over (x - 1)(2x + 5)} \\ {2 \over 1} & = {- 7x - 18 \over 2x^2 + 3x - 5} \\ 2(2x^2 + 3x - 5) & = - 7x - 18 \\ 4x^2 + 6x - 10 & = - 7x - 18 \\ 4x^2 + 6x + 7x - 10 + 18 & = 0 \\ 4x^2 + 13x + 8 & = 0 \\ \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 13 \pm \sqrt{(13)^2 - 4(4)(8)} \over 2(4) } \\ & = { -13 \pm \sqrt{41} \over 8} \\ & \approx -0.82461 \text{ or } -2.4254 \\ & \approx -0.825 \text{ or } -2.43 \end{align}

(i)

(ii)

(iii) Look for the x-intercepts of the graph

\begin{align} \text{From graph, } & x = 0.2 \text{ or } 2.3 \end{align}

(i)

(ii)

(iii) Look for the x-intercepts of the graph

\begin{align} \text{From graph, } & x = -2.6 \text{ or } 0.9 \end{align}

(i)

(ii)

(iii) Look for the x-intercept(s) of the graph

\begin{align} \text{From graph, } & x = -3 \end{align}

(i)

x	-3	-2	-1	0	1	2
y	10	-1	-6	-5	2	15

(ii) Look for the x-intercepts of the graph

\begin{align} \text{From graph, } & x = -2.1 \text{ or } 0.8 \end{align}

(i)

x	-4	-3	-2	-1	0	1	2
y	25	9	1	1	9	25	49

(ii) Look for the x-intercept(s) of the graph

\begin{align} \text{From graph, } & x = -1.5 \end{align}

(a)

\begin{align} y & = (x + 1)(x + 3) \\ y & = x^2 + 3x + x + 3 \\ y & = x^2 + 4x + 3 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (0 + 1)(0 + 3) \\ y & = 3 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x + 1)(x + 3) \\ \\ x & = -1 \text{ or } x = - 3 \phantom{000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {-1 + (-3) \over 2} \\ x & = -2 \\ \\ \text{Let } & x = -2, \\ y & = (-2 + 1)(-2 + 3) \\ y & = -1 \\ \\ \text{Minimum point: } & (-2, -1) \\ \end{align}

(b)

\begin{align} y & = (x - 2)(x + 4) \\ y & = x^2 + 4x - 2x - 8 \\ y & = x^2 + 2x - 8 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (0 - 2)(0 + 4) \\ y & = -8 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x - 2)(x + 4) \\ \\ x & = 2 \text{ or } x = -4 \phantom{000000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {2 + (-4) \over 2} \\ x & = -1 \\ \\ \text{Let } & x = -1, \\ y & = (-1 -2)(-1 + 4) \\ y & = -9 \\ \\ \text{Minimum point: } & (-1, -9) \end{align}

(c)

\begin{align} y & = -(x + 1)(x - 5) \\ y & = -(x^2 - 5x + x - 5) \\ y & = -(x^2 - 4x - 5) \\ y & = - x^2 + 4x + 5 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ \\ \text{Let } & x = 0, \\ y & = -(0 + 1)(0 - 5) \\ y & = 5 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = -(x + 1)(x - 5) \\ \\ x & = -1 \text{ or } x = 5 \phantom{00000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {-1 + 5 \over 2} \\ x & = 2 \\ \\ \text{Let } & x = 2, \\ y & = -(2 + 1)(2 - 5) \\ y & = 9 \\ \\ \text{Maximum point: } & (2, 9) \end{align}

(d)

\begin{align} y & = -(x - 1)(x + 6) \\ y & = -(x^2 + 6x - x - 6) \\ y & = -(x^2 + 5x - 6) \\ y & = - x^2 - 5x + 6 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ \\ \text{Let } & x = 0, \\ y & = -(0 - 1)(0 + 6) \\ y & = 6 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = -(x - 1)(x + 6) \\ \\ x & = 1 \text{ or } x = -6 \phantom{00000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {1 + (-6) \over 2} \\ x & = -2.5 \\ \\ \text{Let } & x = -2.5, \\ y & = -(-2.5 - 1)(-2.5 + 6) \\ y & = 12.25 \\ \\ \text{Maximum point: } & (-2.5, 12.25) \end{align}

(e)

\begin{align} y & = (3 - x)(x + 2) \\ y & = 3x + 6 - x^2 - 2x \\ y & = - x^2 + x + 6 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ \\ \text{Let } & x = 0, \\ y & = (3 - 0)(0 + 2) \\ y & = 6 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (3 - x)(x + 2) \\ \\ x & = 3 \text{ or } x = -2 \phantom{00000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {3 + (-2) \over 2} \\ x & = 0.5 \\ \\ \text{Let } & x = 0.5, \\ y & = (3 - 0.5)(0.5 + 2) \\ y & = 6.25 \\ \\ \text{Maximum point: } & (0.5, 6.25) \end{align}

(f)

\begin{align} y & = (2 - x)(4 - x) \\ y & = 8 - 2x - 4x + x^2 \\ y & = x^2 - 6x + 8 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (2 - 0)(4 - 0) \\ y & = 8 \phantom{0000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (2 - x)(4 - x) \\ \\ x & = 2 \text{ or } x = 4 \phantom{000000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {2 + 4 \over 2} \\ x & = 3 \\ \\ \text{Let } & x = 3, \\ y & = (2 - 3)(4 - 3) \\ y & = -1 \\ \\ \text{Minimum point: } & (3, -1) \end{align}

(a)

\begin{align} y & = x^2 - 4 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = (x - 0)^2 - 4 \\ \\ \text{Minimum point: } & (0, -4) \\ \\ \text{Line of symmetry: } & x = 0 \\ \\ \text{Let } & x = 0, \\ y & = (0)^2 - 4 \\ y & = - 4 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(b)

\begin{align} y & = -x^2 + 6 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ y & = -(x - 0)^2 + 6 \\ \\ \text{Maximum point: } & (0, 6) \\ \\ \text{Line of symmetry: } & x = 0 \\ \\ \text{Let } & x = 0, \\ y & = -(0)^2 + 6 \\ y & = 6 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(c)

\begin{align} y & = (x - 3)^2 - 2 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Minimum point: } & (3, -2) \\ \\ \text{Line of symmetry: } & x = 3 \\ \\ \text{Let } & x = 0, \\ y & = (0 - 3)^2 - 2 \\ y & = 7 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(d)

\begin{align} y & = (x + 1)^2 - 3 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = [x - (-1)]^2 - 3 \\ \\ \text{Minimum point: } & (-1, -3) \\ \\ \text{Line of symmetry: } & x = -1 \\ \\ \text{Let } & x = 0, \\ y & = (0 + 1)^2 - 3 \\ y & = -2 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(e)

\begin{align} y & = -(x + 2)^2 + 3 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ y & = -[x - (-2)]^2 + 3 \\ \\ \text{Maximum point: } & (-2, 3) \\ \\ \text{Line of symmetry: } & x = -2 \\ \\ \text{Let } & x = 0, \\ y & = -(0 + 2)^2 + 3 \\ y & = -1 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(f)

\begin{align} y & = -(x - 4)^2 + 5 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ \\ \text{Maximum point: } & (4, 5) \\ \\ \text{Line of symmetry: } & x = 4 \\ \\ \text{Let } & x = 0, \\ y & = - (0 - 4)^2 + 5 \\ y & = -11 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(i)

\begin{align} x^2 + {3 \over 4}x & = x\left(x + {3 \over 4}\right) \end{align}

(ii)

\begin{align} y & = x^2 + {3 \over 4}x \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = x \left(x + {3 \over 4}\right) \\ \\ \text{Let } & x = 0, \\ y & = (0)\left(0 + {3 \over 4}\right) \\ y & = 0 \phantom{000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = x \left(x + {3 \over 4}\right) \\ \\ x & = 0 \text{ or } x = -{3 \over 4} \phantom{000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = { 0 + \left(-{3 \over 4}\right) \over 2} \\ x & = -{3 \over 8} \\ \\ \text{Let } & x = -{3 \over 8}, \\ y & = \left(-{3 \over 8}\right) \left(-{3 \over 8} + {3 \over 4}\right) \\ y & = -{9 \over 64} \\ \\ \text{Minimum point: } & \left(-{3 \over 8}, -{9 \over 64}\right) \end{align}

(i)

\begin{align} x^2 + x - 6 & = (x - 2)(x + 3) \end{align}

(ii)

\begin{align} y & = x^2 + x - 6 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Let } & x = 0, \\ y & = (0)^2 + (0) - 6 \\ y & = -6 \phantom{000000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = x^2 + x - 6 \\ 0 & = (x - 2)(x + 3) \phantom{000000000} [\text{Use result from (i)}] \\ \\ x & = 2 \text{ or } x = -3 \phantom{00000000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = { 2 + (-3) \over 2} \\ x & = -0.5 \\ \\ \text{Let } & x = -0.5, \\ y & = (-0.5)^2 + (-0.5) - 6 \\ y & = -6.25 \\ \\ \text{Minimum point: } & (-0.5, -6.25) \end{align}

(a)(i)

\begin{align} y & = -x^2 - 7x - 15 \\ & = -(x^2 + 7x) - 15 \\ & = - \left[ x^2 + 7x + \left(7 \over 2\right)^2 - \left(7 \over 2\right)^2 \right] - 15 \phantom{00000000} [\text{Complete the square}] \\ & = - ( x^2 + 7x + 3.5^2 - 3.5^2 ) - 15 \\ & = - [(x + 3.5)^2 - 12.25] - 15 \\ & = - (x + 3.5)^2 + 12.25 - 15 \\ & = - (x + 3.5)^2 - 2.75 \end{align}

(a)(ii)

\begin{align} y & = - x^2 - 7x - 15 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ y & = - (x + 3.5)^2 - 2.75 \\ y & = - [x - (-3.5)]^2 - 2.75 \\ \\ \text{Maximum point: } & (-3.5, -2.75) \\ \\ \text{Let } & x = 0, \\ y & = - (0 + 3.5)^2 - 2.75 \\ y & = -15 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(b)(i)

\begin{align} y & = x^2 - 3x + 4 \\ & = x^2 - 3x + \left(3 \over 2\right)^2 - \left(3 \over 2\right)^2 + 4 \phantom{00000000} [\text{Complete the square}] \\ & = x^2 - 3x + 1.5^2 - 1.5^2 + 4 \\ & = (x - 1.5)^2 - 2.25 + 4 \\ & = (x - 1.5)^2 + 1.75 \end{align}

(b)(ii)

\begin{align} y & = x^2 - 3x + 4 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = (x - 1.5)^2 + 1.75 \\ \\ \text{Minimum point: } & (1.5, 1.75) \\ \\ \text{Let } & x = 0, \\ y & = (0 - 1.5)^2 + 1.75 \\ y & = 4 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(a)(i)

\begin{align} y & = - x^2 - 10x - 25 \\ & = -(x^2 + 10x) - 25 \\ & = - \left[ x^2 + 10x + \left(10 \over 2\right)^2 - \left(10 \over 2\right)^2 \right] - 25 \phantom{00000000} [\text{Complete the square}] \\ & = - ( x^2 + 10x + 5^2 - 5^2 ) - 25 \\ & = - [(x + 5)^2 - 25] - 25 \\ & = - (x + 5)^2 + 25 - 25 \\ & = - (x + 5)^2 \end{align}

(a)(ii)

\begin{align} y & = - x^2 - 10x - 25 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ y & = - (x + 5)^2 \\ y & = - [x - (-5)]^2 + 0 \\ \\ \text{Maximum point: } & (-5, 0) \\ \\ \text{Let } & x = 0, \\ y & = - (0 + 5)^2 \\ y & = -25 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(b)(i)

\begin{align} y & = x^2 - 8x + 16 \\ & = x^2 - 8x + \left(8 \over 2\right)^2 - \left(8 \over 2\right)^2 + 16 \phantom{00000000} [\text{Complete the square}] \\ & = x^2 - 8x + 4^2 - 4^2 + 16 \\ & = (x - 4)^2 - 16 + 16 \\ & = (x - 4)^2 \end{align}

(b)(ii)

\begin{align} y & = x^2 - 8x + 16 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = (x - 4)^2 \\ y & = (x - 4)^2 + 0 \\ \\ \text{Minimum point: } & (4, 0) \\ \\ \text{Let } & x = 0, \\ y & = (0 - 4)^2 \\ y & = 16 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(a)(i)

\begin{align} y & = -x^2 + 6x - 6 \\ & = -(x^2 - 6x) - 6 \\ & = - \left[ x^2 - 6x + \left(6 \over 2\right)^2 - \left(6 \over 2\right)^2 \right] - 6 \phantom{00000000} [\text{Complete the square}] \\ & = - (x^2 - 6x + 3^2 - 3^2) - 6 \\ & = - [ (x - 3)^2 - 9 ] - 6 \\ & = - (x - 3)^2 + 9 - 6 \\ & = - (x - 3)^2 + 3 \end{align}

(a)(ii)

\begin{align} y & = - x^2 + 6x - 6 \phantom{0000000} [\text{Maximum curve } (\cap)] \\ y & = -(x - 3)^2 + 3 \\ \\ \text{Maximum point: } & (3, 3) \\ \\ \text{Let } & x = 0, \\ y & = -(0 - 3)^2 + 3 \\ y & = -6 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(b)(i)

\begin{align} y & = x^2 - 8x + 5 \\ & = x^2 - 8x + \left(8 \over 2\right)^2 - \left(8 \over 2\right)^2 + 5 \phantom{00000000} [\text{Complete the square}] \\ & = x^2 - 8x + 4^2 - 4^2 + 5 \\ & = (x - 4)^2 - 16 + 5 \\ & = (x - 4)^2 - 11 \end{align}

(b)(ii)

\begin{align} y & = x^2 - 8x + 5 \phantom{0000000} [\text{Minimum curve } (\cup)] \\ y & = (x - 4)^2 - 11 \\ \\ \text{Minimum point: } & (4, -11) \\ \\ \text{Let } & x = 0, \\ y & = (0 - 4)^2 - 11 \\ y & = 5 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(i)

\begin{align} \text{Minimum point: } & \left(-{1 \over 2}, {3 \over 4} \right) \\ \\ y = \left[ x - \left(-{1 \over 2}\right) \right]^2 & + {3 \over 4} \\ \\ \\ h = -{1 \over 2}, k & = {3 \over 4} \end{align}

(ii)

\begin{align} y = \left[ x - \left(-{1 \over 2}\right) \right]^2 & + {3 \over 4} \phantom{0000000000} [\text{Minimum curve } (\cup)] \\ \\ \text{Minimum point: } & \left(-{1 \over 2}, {3 \over 4} \right) \\ \\ \text{Let } & x = 0, \\ y & = \left[0 - \left(-{1 \over 2}\right) \right]^2 + {3 \over 4} \\ y & = 1 \phantom{000000000000000000} [ y \text{-intercept}] \end{align}

(i)

\begin{align} \text{Perimeter} & = 2 \times \text{Length} + 2 \times \text{Breadth} \\ 112 & = 2 \times \text{Length} + 2 \times x \\ 112 & = 2 \times \text{Length} + 2x \\ 112 - 2x & = 2 \times \text{Length} \\ \\ \text{Length} & = {1 \over 2}(112 - 2x) \\ & = (56 - x) \text{ cm} \end{align}

(ii)

\begin{align} \text{Area} & = \text{Length} \times \text{Breadth} \\ 597 & = (56 - x) \times x \\ 597 & = x(56 - x) \\ 597 & = 56x - x^2 \\ x^2 - 56x + 597 & = 0 \phantom{0000} \text{ (Shown)} \end{align}

(iii)

\begin{align} x^2 & - 56x + 597 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-56) \pm \sqrt{(-56)^2 - 4(1)(597)} \over 2(1)} \\ & = {56 \pm \sqrt{748} \over 2} \\ & \approx 41.674794 \text{ or } 14.325206 \\ & \approx 41.67 \text{ or } 14.33 \text{ (2 d.p.)} \end{align}

(iv) Place the posters with their breadth along the wall.

\begin{align} \text{Breadth, } x & = {56 - \sqrt{748} \over 2} \text{ cm} \\ \\ 4 \text{ m} & = 400 \text{ cm} \\ \\ \text{Number of poster widths} & = {400 \over {56 - \sqrt{748} \over 2}} \\ & \approx 27.922810 \\ \\ \therefore \text{Max. posters} & = 27 \end{align}

(i)

\begin{align} x \text{ mins} & \rightarrow 60 \text{ pages} \\ 1 \text{ min} & \rightarrow {60 \over x} \text{ pages} \end{align}

(ii)

\begin{align} (x + 2) \text{ mins} & \rightarrow 60 \text{ pages} \\ 1 \text{ min} & \rightarrow {60 \over x+2} \text{ pages} \end{align}

(iii)

\begin{align} {60 \over x} + {60 \over x + 2} & = 144 \\ {60(x + 2) \over x(x + 2)} + {60x \over x(x + 2)} & = 144 \\ {60(x + 2) + 60x \over x(x + 2)} & = 144 \\ {60x + 120 + 60x \over x^2 + 2x} & = 144 \\ {120x + 120 \over x^2 + 2x} & = {144 \over 1} \\ 120x + 120 & = 144(x^2 + 2x) \\ 120x + 120 & = 144x^2 + 288x \\ 0 & = 144x^2 + 288x - 120x -120 \\ 0 & = 144x^2 + 168x - 120 \\ 0 & = 6x^2 + 7x - 5 \phantom{00} \text{ (Shown)} \phantom{000000} [\text{Divide by 24}] \end{align}

(iv)

\begin{align} 0 & = 6x^2 + 7x - 5 \\ 0 & = (2x - 1)(3x + 5) \end{align} \begin{align} 2x - 1 & = 0 && \text{ or } & 3x + 5 & = 0 \\ 2x & = 1 &&& 3x & = -5 \\ x & = {1 \over 2} &&& x & = -{5 \over 3} \end{align}

(v) Since x is a time, use the positive solution from (iv).

\begin{align} \text{Pages by Printer B in 1 min} & = {60 \over x + 2} \\ & = {60 \over {1 \over 2} + 2} \\ & = 24 \\ \\ \text{Time taken} & = {144 \over 24} \\ & = 6 \text{ mins} \end{align}

(i)

\begin{align} \$ x & \rightarrow 1 \text{ kg} \\ \$ 1 & \rightarrow {1 \over x} \text{ kg} \\ \$ 600 & \rightarrow {600 \over x} \text{ kg} \end{align}

(ii)

\begin{align} \$ (x - 0.4) & \rightarrow 1 \text{ kg} \\ \$ 1 & \rightarrow {1 \over x - 0.4} \text{ kg} \\ \$ 600 & \rightarrow {600 \over x - 0.4} \text{ kg} \end{align}

(iii) Since brand B is cheaper, the food catering company will obtain more rice from brand B

\begin{align} {600 \over x - 0.4} - {600 \over x} & = 45 \\ {600x \over x(x - 0.4)} - {600(x - 0.4) \over x(x - 0.4)} & = 45 \\ {600x - 600(x - 0.4) \over x(x - 0.4)} & = 45 \\ {600x - 600x + 240 \over x^2 - 0.4x} & = 45 \\ {240 \over x^2 - 0.4x} & = {45 \over 1} \\ 240 & = 45(x^2 - 0.4x) \\ 240 & = 45x^2 - 18x \\ 0 & = 45x^2 - 18x - 240 \\ 0 & = 15x^2 - 6x - 80 \phantom{00} \text{ (Shown)} \phantom{000000} [\text{Divide by 3}] \end{align}

(iv)

\begin{align} 0 & = 15x^2 - 6x - 80 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-6) \pm \sqrt{(-6)^2 - 4(15)(-80)} \over 2(15)} \\ & = {6 \pm \sqrt{4836} \over 30} \\ & \approx 2.518045 \text{ or } -2.118045 \\ \\ \\ \text{Price of 1 kg from B} & = x - 0.4 \\ & = {6 + \sqrt{4836} \over 30} - 0.4 \\ & \approx 2.118045 \\ & \approx \$ 2.12 \end{align}

\begin{align} h & = -t^2 + 5t + 10 \phantom{00000000000000000000000.} [\text{Maximum curve } \cap] \\ h & = -(t^2 - 5t) + 10 \\ h & = -\left[ t^2 - 5t + \left(5 \over 2\right)^2 - \left(5 \over 2\right)^2 \right] + 10 \phantom{000000} [\text{Complete the square}] \\ h & = - ( t^2 - 5t + 2.5^2 - 2.5^2 ) + 10 \\ h & = - [(t - 2.5)^2 - 6.25] + 10 \\ h & = -(t - 2.5)^2 + 6.25 + 10 \\ h & = -(t - 2.5)^2 + 16.25 \\ \\ \text{Max. point: } & (2.5, 16.25) \\ \\ \text{Max. height} & = 16.25 \text{ m} \\ \\ \\ \text{When } & t = 0, \\ h & = -(0 - 2.5)^2 + 16.25 \\ & = 10 \text{ m} \\ \\ \text{Initial height} & = 10 \text{ m} \\ \\ \\ \text{Height difference} & = 16.25 - 10 \\ & = 6.25 \text{ m} \end{align}

(i)

\begin{align} \text{Area of triangle} & = {1 \over 2} \times BC \times AP \\ & = {1 \over 2} \times 2x \times (x + 5) \\ & = x(x + 5) \\ & = x^2 + 5x \\ \\ \text{Area of base (square)} & = 2x \times 2x \\ & = 4x^2 \\ \\ \text{Total surface area} & = 4x^2 + 4(x^2 + 5x) \\ & = 4x^2 + 4x^2 + 20x \\ & = (8x^2 + 20x) \text{ cm}^2 \end{align}

(ii)

\begin{align} \text{Total surface area} & = 8x^2 + 20x \\ 50 & = 8x^2 + 20x \\ 0 & = 8x^2 + 20x - 50 \\ 0 & = 4x^2 + 10x - 25 \phantom{00} \text{ (Shown)} \phantom{000000} [\text{Divide by 2}] \end{align}

(iii)

\begin{align} 0 & = 4x^2 + 10x - 25 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-10 \pm \sqrt{(10)^2 - 4(4)(-25)} \over 2(4)} \\ & = {-10 \pm \sqrt{500} \over 8} \\ & \approx 1.545085 \text{ or } -4.045085 \\ & \approx 1.55 \text{ or } -4.05 \end{align}

(iv)

\begin{align} BC > 0, \text{ thus the negative root must be rejected} \end{align}

(v)

\begin{align} \text{Floor area of model} & = 4x^2 \phantom{00000000} [\text{Area of square base}] \\ & \approx 4(1.55)^2 \phantom{0000} [\text{Using the rounded value from (iii)}] \\ & = 9.61 \text{ cm}^2 \\ \\ 1 & : 100 \\ 1 \text{ cm} & : 100 \text{ cm} \\ 1 \text{ cm} & : 1 \text{ m} \\ 1^2 \text{ cm}^2 & : 1^2 \text{ m}^2 \\ 1 \text{ cm}^2 & : 1 \text{ m}^2 \\ 9.61 \text{ cm}^2 & : 9.61 \text{ m}^2 \\ \\ \text{Actual floor area} & \approx 9.61 \text{ m}^2 \end{align}

(i)

\begin{align} \text{Time} & = {\text{Distance} \over \text{Speed}} \\ \\ \text{Time taken by Weiming} & = {2 \over x} + {10 - 2 \over x + 1} \\ & = \left({2 \over x} + {8 \over x + 1}\right) \text{ hours} \end{align}

(ii)

\begin{align} \text{Time taken by Yi Hao} & = {10 \over x} \text{ hours} \\ \\ \text{30 mins} & = {1 \over 2} \text{ hour} \\ \\ {10 \over x} - \left({2 \over x} + {8 \over x + 1}\right) & = {1 \over 2} \\ {10 \over x} - {2 \over x} - {8 \over x + 1} & = {1 \over 2} \\ {8 \over x} - {8 \over x + 1} & = {1 \over 2} \\ {8(x + 1) \over x(x + 1)} - {8x \over x(x + 1)} & = {1 \over 2} \\ {8(x + 1) - 8x \over x(x + 1)} & = {1 \over 2} \\ {8x + 8 - 8x \over x^2 + x} & = {1 \over 2} \\ {8 \over x^2 + x} & = {1 \over 2} \\ 2(8) & = x^2 + x \\ 16 & = x^2 + x \\ 0 & = x^2 + x - 16 \phantom{00} \text{ (Shown)} \end{align}

(iii)

\begin{align} 0 & = x^2 + x - 16 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-1 \pm \sqrt{(1)^2 - 4(1)(-16)} \over 2(1)} \\ & = {-1 \pm \sqrt{65} \over 2} \\ & \approx 3.531129 \text{ or } -4.531129 \\ & \approx 3.53 \text{ or } -4.53 \\ \\ \\ \text{Since } & x > 0, \text{ the negative root must be rejected} \end{align}

(iv)

\begin{align} \text{Let } x & = {-1 + \sqrt{65} \over 2} \\ \\ \text{Time taken by Weiming} & = {2 \over x} + {8 \over x + 1} \\ & \approx 2.331956 \text{ hours} \\ & \approx 2 \text{ hours } 19.917333 \text{ mins} \\ & \approx 2 \text{ hours } 20 \text{ mins} \\ \\ 8:00 + 2:20 & = 10:20 \\ \\ \\ \text{Weiming finished} & \text{ the hike at } 10:20 \text{ am} \end{align}

(i)

\begin{align} \text{Time} & = {\text{Distance} \over \text{Speed}} \\ \\ \text{Time taken} & = \underbrace{{700 \over x - 30}}_\text{By bus} + \underbrace{700 \over x}_\text{By car} \\ 20 & = {700x \over x(x - 30)} + {700(x - 30) \over x(x - 30)} \\ 20 & = {700x + 700(x - 30) \over x(x - 30)} \\ 20 & = {700x + 700x - 21 \phantom{.} 000 \over x^2 - 30x} \\ {20 \over 1} & = {1400x - 21 \phantom{.} 000 \over x^2 - 30x} \\ 20(x^2 - 30x) & = 1400x - 21 \phantom{.} 000 \\ 20x^2 - 600x & = 1400x - 21 \phantom{.} 000 \\ 20x^2 - 600x - 1400x + 21 \phantom{.} 000 & = 0 \\ 20x^2 - 2000x + 21 \phantom{.} 000 & = 0 \\ x^2 - 100x + 1050 & = 0 \phantom{00} \text{ (Shown)} \phantom{000000} [\text{Divide by 20}] \end{align}

(ii)

\begin{align} x^2 & - 100x + 1050 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-100) \pm \sqrt{(-100)^2 - 4(1)(1050)} \over 2(1)} \\ & = {100 \pm \sqrt{5800} \over 2} \\ & \approx 88.078866 \text{ or } 11.921134 \\ & \approx 88.08 \text{ or } 11.92 \text{ (2 d.p.)} \end{align}

(iii) The smaller root gives a negative bus speed, so use the larger root.

\begin{align} \text{Time taken for return journey by car} & = {700 \over x} \\ & = {700 \over {100 + \sqrt{5800} \over 2}} \\ & \approx 7.947423 \\ & \approx 7.95 \text{ hours} \end{align}

(i)

\begin{align} x \phantom{0} l & \rightarrow 1 \text{ min} \\ 1 \phantom{0} l & \rightarrow {1 \over x} \text{ min} \\ 1500 \phantom{0} l & \rightarrow {1500 \over x} \text{ mins} \end{align}

(ii)

\begin{align} (x + 50) \phantom{0} l & \rightarrow 1 \text{ min} \\ 1 \phantom{0} l & \rightarrow {1 \over x + 50} \text{ min} \\ 1500 \phantom{0} l & \rightarrow {1500 \over x + 50} \text{ mins} \end{align}

(iii)

\begin{align} 30 \text{ s} & = {1 \over 2} \text{ min} \\ \\ {1500 \over x} - {1500 \over x + 50} & = {1 \over 2} \\ {1500(x + 50) \over x(x + 50)} - {1500x \over x(x + 50)} & = {1 \over 2} \\ {1500(x + 50) - 1500x \over x(x + 50)} & = {1 \over 2} \\ {1500x + 75 \phantom{.} 000 - 1500x \over x^2 + 50x} & = {1 \over 2} \\ {75 \phantom{.} 000 \over x^2 + 50x} & = {1 \over 2} \\ 2(75 \phantom{.} 000) & = x^2 + 50x \\ 150 \phantom{.} 000 & = x^2 + 50x \\ 0 & = x^2 + 50x - 150 \phantom{.} 000 \phantom{00} \text{ (Shown)} \end{align}

(iv)

\begin{align} 0 & = x^2 + 50x - 150 \phantom{.} 000 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-50 \pm \sqrt{(50)^2 - 4(1)(-150 \phantom{.} 000)} \over 2(1)} \\ & = {-50 \pm \sqrt{602 \phantom{.} 500} \over 2} \\ & \approx 363.104367 \text{ or } -413.104367 \\ & \approx 363.10 \text{ or } -413.10 \text{ (2 d.p.)} \end{align}

(v)

\begin{align} \text{Time taken by Pump B} & = {1500 \over x + 50} \\ & = {1500 \over {-50 + \sqrt{602500} \over 2} + 50} \\ & \approx 3.631044 \text{ mins} \\ & \approx 3 \text{ mins } 37.862620 \text{ s} \\ & \approx 3 \text{ mins } 38 \text{ s} \end{align}

(i)

\begin{align} \text{S\$}x & \rightarrow \text{US\$}1 \\ \text{S\$}1 & \rightarrow \text{US\$} {1 \over x} \\ \text{S\$}2000 & \rightarrow \text{US\$} {2000 \over x} \end{align}

(ii)

\begin{align} \text{S\$}(x + 0.05) & \rightarrow \text{US\$}1 \\ \text{S\$}1 & \rightarrow \text{US\$} {1 \over x + 0.05} \\ \text{S\$}1000 & \rightarrow \text{US\$} {1000 \over x + 0.05} \end{align}

(iii)

\begin{align} {2000 \over x} + {1000 \over x + 0.05} & = 2370 \\ {2000(x + 0.05) \over x(x + 0.05)} + {1000x \over x(x + 0.05)} & = 2370 \\ {2000(x + 0.05) + 1000x \over x(x + 0.05)} & = 2370 \\ {2000x + 100 + 1000x \over x^2 + 0.05x} & = 2370 \\ {3000x + 100 \over x^2 + 0.05x} & = {2370 \over 1} \\ 3000x + 100 & = 2370(x^2 + 0.05x) \\ 3000x + 100 & = 2370x^2 + 118.5x \\ 0 & = 2370x^2 + 118.5x - 3000x - 100 \\ 0 & = 2370x^2 - 2881.5x - 100 \\ 0 & = 237x^2 - 288.15x - 10 \phantom{0} \text{ (Shown)} \phantom{000000} [\text{Divide by 10}] \end{align}

(iv)

\begin{align} 0 & = 237x^2 - 288.15x - 10 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-288.15) \pm \sqrt{(-288.15)^2 - 4(237)(-10)} \over 2(237)} \\ & = {288.15 \pm \sqrt{92 \phantom{.} 510. 422 \phantom{.} 5 } \over 474} \\ & \approx 1.249589 \text{ or } -0.033766 \\ & \approx 1.2496 \text{ or } -0.0338 \text{ (4 d.p.)} \end{align}

(v)

\begin{align} x + 0.05 & = {288.15 + \sqrt{92510.4225} \over 474} + 0.05 \\ & \approx 1.299589 \\ & \approx 1.30 \\ \\ \text{Exchange rate: } & \text{US\$}1 = \text{S\$}1.30 \end{align}

Assume that the aircraft maintains a constant airspeed of 165 km/h on both legs, the wind acts along the flight route, and take-off, landing and turnaround times are ignored.

\begin{align} \text{Time} & = { \text{Distance} \over \text{Speed} } \\ \\ \text{Time taken (with wind)} & = { 450 \over 165 + x} \\ \\ \text{Time taken (against wind)} & = {450 \over 165 - x} \\ \\ 5 \text{ hours } 30 \text{ mins} & = 5{1 \over 2} \text{ hours} \\ & = {11 \over 2} \text{ hours} \\ \\ {450 \over 165 + x} + {450 \over 165 - x} & = {11 \over 2} \\ {450(165 - x) \over (165 + x)(165 - x)} + {450(165 + x) \over (165 + x)(165 - x)} & = {11 \over 2} \\ {450(165 - x) + 450(165 + x) \over (165 + x)(165 - x)} & = {11 \over 2} \\ {74 \phantom{.} 250 - 450x + 74 \phantom{.} 250 + 450x \over (165)^2 - (x)^2} & = {11 \over 2} \\ {148 \phantom{.} 500 \over 27 \phantom{.} 225 - x^2} & = {11 \over 2} \\ 2(148 \phantom{.} 500) & = 11(27 \phantom{.} 225 - x^2) \\ 297 \phantom{.} 000 & = 299 \phantom{.} 475 - 11x^2 \\ 11x^2 & = 299 \phantom{.} 475 - 297 \phantom{.} 000 \\ 11x^2 & = 2475 \\ x^2 & = {2475 \over 11} \\ x^2 & = 225 \\ x & = \pm \sqrt{225} \\ x & = \pm 15 \\ \\ \\ \therefore \text{Speed of wind} & = 15 \text{ km/h} \end{align}

(i)

(ii)(a)

\begin{align} \text{Max. height} & = 202.5 \text{ cm} \end{align}

(ii)(b)

\begin{align} \text{Horizontal distance} & = 5.6 \text{ m} \end{align}

(iii)

\begin{align} \text{From graph, } & \text{balloon reaches ground (}y = 0) \text{ when } x = 6.4 \\ \\ \therefore t & = 6.4 \end{align}

(a)

\begin{align} x^2 + 8x + 5 & = 0 \\ x^2 + 8x + \left(8 \over 2\right)^2 - \left(8 \over 2\right)^2 + 5 & = 0 \\ x^2 + 8x + 4^2 - 4^2 + 5 & = 0 \\ (x + 4)^2 - 16 + 5 & = 0 \\ (x + 4)^2 - 11 & = 0 \\ (x + 4)^2 & = 11 \\ x + 4 & = \pm \sqrt{11} \end{align} \begin{align} x + 4 & = \sqrt{11} && \text{ or } & x + 4 & = - \sqrt{11} \\ x & = \sqrt{11} - 4 &&& x & = - \sqrt{11} - 4 \\ x & \approx -0.683 &&& x & \approx -7.32 \end{align}

(b)

\begin{align} x^2 + 7x - 3 & = 0 \\ x^2 + 7x + \left(7 \over 2\right)^2 - \left(7 \over 2\right)^2 - 3 & = 0 \\ x^2 + 7x + 3.5^2 - 3.5^2 - 3 & = 0 \\ (x + 3.5)^2 - 12.25 - 3 & = 0 \\ (x + 3.5)^2 - 15.25 & = 0 \\ (x + 3.5)^2 & = 15.25 \\ x + 3.5 & = \pm \sqrt{15.25} \end{align} \begin{align} x + 3.5 & = \sqrt{15.25} && \text{ or } & x + 3.5 & = - \sqrt{15.25} \\ x & = \sqrt{15.25} - 3.5 &&& x & = - \sqrt{15.25} - 3.5 \\ x & \approx 0.405 &&& x & \approx -7.41 \end{align}

(c)

\begin{align} x^2 - 11x - 7 & = 0 \\ x^2 - 11x + \left(11 \over 2\right)^2 - \left(11 \over 2\right)^2 - 7 & = 0 \\ x^2 - 11x + 5.5^2 - 5.5^2 - 7 & = 0 \\ (x - 5.5)^2 - 30.25 - 7 & = 0 \\ (x - 5.5)^2 - 37.25 & = 0 \\ (x - 5.5)^2 & = 37.25 \\ x - 5.5 & = \pm \sqrt{37.25} \end{align} \begin{align} x - 5.5 & = \sqrt{37.25} && \text{ or } & x - 5.5 & = - \sqrt{37.25} \\ x & = \sqrt{37.25} + 5.5 &&& x & = - \sqrt{37.25} + 5.5 \\ x & \approx 11.6 &&& x & \approx -0.603 \end{align}

(d)

\begin{align} x^2 + 1.2x & = 1 \\ x^2 + 1.2x + \left(1.2 \over 2\right)^2 - \left(1.2 \over 2\right)^2 & = 1 \\ x^2 + 1.2x + 0.6^2 - 0.6^2 & = 1 \\ (x + 0.6)^2 - 0.36 & = 1 \\ (x + 0.6)^2 & = 1 + 0.36 \\ (x + 0.6)^2 & = 1.36 \\ x + 0.6 & = \pm \sqrt{1.36} \end{align} \begin{align} x + 0.6 & = \sqrt{1.36} && \text{ or } & x + 0.6 & = - \sqrt{1.36} \\ x & = \sqrt{1.36} - 0.6 &&& x & = -\sqrt{1.36} - 0.6 \\ x & \approx 0.566 &&& x & \approx -1.77 \end{align}

(a)

\begin{align} 2x^2 & + 6x + 1 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-6 \pm \sqrt{(6)^2 - 4(2)(1)} \over 2(2)} \\ & = {-6 \pm \sqrt{28} \over 4} \\ & \approx -0.177124 \text{ or } -2.822876 \\ & \approx -0.177 \text{ or } -2.82 \end{align}

(b)

\begin{align} 3x^2 & - 7x - 2 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-7) \pm \sqrt{(-7)^2 - 4(3)(-2)} \over 2(3)} \\ & = {7 \pm \sqrt{73} \over 6} \\ & \approx 2.590667 \text{ or } -0.257334 \\ & \approx 2.59 \text{ or } -0.257 \end{align}

(c)

\begin{align} -4x^2 & + x + 5 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-1 \pm \sqrt{(1)^2 - 4(-4)(5)} \over 2(-4)} \\ & = {-1 \pm \sqrt{81} \over -8} \\ & = -1 \text{ or } {5 \over 4} \end{align}

(d)

\begin{align} 3x^2 & = 5x + 1 \\ 3x^2 - 5x - 1 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-5) \pm \sqrt{(-5)^2 - 4(3)(-1)} \over 2(3)} \\ & = {5 \pm \sqrt{37} \over 6} \\ & \approx 1.847127 \text{ or } -0.180460 \\ & \approx 1.85 \text{ or } - 0.180 \end{align}

(a)

\begin{align} (x - 3)^2 & = {4 \over 25} \\ x - 3 & = \pm \sqrt{4 \over 25} \\ x - 3 & = \pm {2 \over 5} \end{align} \begin{align} x - 3 & = {2 \over 5} && \text{ or } & x - 3 & = -{2 \over 5} \\ x & = {2 \over 5} + 3 &&& x & = -{2 \over 5} + 3 \\ x & = {17 \over 5} &&& x & = {13 \over 5} \end{align}

(b)

\begin{align} (4 - x)^2 & = 12 \\ 4 - x & = \pm \sqrt{12} \end{align} \begin{align} 4 - x & = \sqrt{12} && \text{ or } & 4 - x & = -\sqrt{12} \\ 4 - \sqrt{12} & = x &&& 4 + \sqrt{12} & = x \\ 0.536 & \approx x &&& 7.46 & \approx x \end{align}

(c)

\begin{align} (x - 1)(x + 3) & = 9 \\ x^2 + 3x - x - 3 & = 9 \\ x^2 + 2x - 3 & = 9 \\ x^2 + 2x - 3 - 9 & = 0 \\ x^2 + 2x - 12 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-2 \pm \sqrt{(2)^2 - 4(1)(-12)} \over 2(1)} \\ & = {-2 \pm \sqrt{52} \over 2} \\ & \approx 2.605551 \text{ or } -4.605551 \\ & \approx 2.61 \text{ or } -4.61 \end{align}

(d)

\begin{align} x(x + 4) & = 17 \\ x^2 + 4x & = 17 \\ x^2 + 4x - 17 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-4 \pm \sqrt{(4)^2 - 4(1)(-17)} \over 2(1)} \\ & = {-4 \pm \sqrt{84} \over 2} \\ & \approx 2.582576 \text{ or } -6.582576 \\ & \approx 2.58 \text{ or } -6.58 \end{align}

(i)

\begin{align} 2x^2 & - 7x + 4 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-7) \pm \sqrt{(-7)^2 - 4(2)(4)} \over 2(2)} \\ & = {7 \pm \sqrt{ 17} \over 4} \\ & \approx 2.780776 \text{ or } 0.719224 \\ & \approx 2.78 \text{ or } 0.72 \text{ (2 d.p.)} \end{align}

(ii)

\begin{align} 2x^2 & - 7x + 4 = 0 \\ \\ \text{Replace } & x \text{ by } y - 1, \\ 2(y - 1)^2 & - 7(y - 1) + 4 = 0 \\ \\ \\ \text{From (i), } x & = {7 \pm \sqrt{17} \over 4} \\ y - 1 & = {7 \pm \sqrt{17} \over 4} \end{align} \begin{align} y - 1 & = {7 + \sqrt{17} \over 4} && \text{ or } & y - 1 & = {7 - \sqrt{17} \over 4} \\ y & = 1 + {7 + \sqrt{17} \over 4} &&& y & = 1 + {7 - \sqrt{17} \over 4} \\ y & \approx 3.780776 &&& y & \approx 1.719224 \\ y & \approx 3.78 &&& y & \approx 1.72 \end{align}

(a)

\begin{align} x & = 2 && \text{ or } & x & = {6 \over 7} \\ x - 2 & = 0 &&& 7x & = 6 \\ & &&& 7x - 6 & = 0 \end{align} \begin{align} (x - 2)(7x - 6) & = 0 \\ 7x^2 - 6x - 14x + 12 & = 0 \\ 7x^2 - 20x + 12 & = 0 \end{align}

(b)

\begin{align} x & = -{1 \over 2} && \text{ or } & x & = -{2 \over 3} \\ 2x & = -1 &&& 3x & = -2 \\ 2x + 1 & =0 &&& 3x + 2 & = 0 \end{align} \begin{align} (2x + 1)(3x + 2) & = 0 \\ 6x^2 + 4x + 3x + 2 & = 0 \\ 6x^2 + 7x + 2 & = 0 \end{align}

(a)

\begin{align} {x - 1 \over 1} & = {5 \over x + 7} \\ (x + 7)(x - 1) & = 5 \\ x^2 - x + 7x - 7 & = 5 \\ x^2 + 6x - 7 & = 5 \\ x^2 + 6x - 7 - 5 & = 0 \\ x^2 + 6x - 12 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-6 \pm \sqrt{(6)^2 - 4(1)(-12)} \over 2(1)} \\ & = {-6 \pm \sqrt{84} \over 2} \\ & \approx 1.582576 \text{ or } -7.582576 \\ & \approx 1.58 \text{ or } -7.58 \end{align}

(b)

\begin{align} {x - 1 \over x + 4} & = {2x \over x - 3} \\ (x - 3)(x - 1) & = 2x(x + 4) \\ x^2 - x - 3x + 3 & = 2x^2 + 8x \\ x^2 - 4x + 3 & = 2x^2 + 8x \\ 0 & = 2x^2 - x^2 + 8x + 4x - 3 \\ 0 & = x^2 + 12x - 3 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-12 \pm \sqrt{(12)^2 - 4(1)(-3)} \over 2(1)} \\ & = {-12 \pm \sqrt{156} \over 2} \\ & \approx 0.244998 \text{ or } -12.244998 \\ & \approx 0.245 \text{ or } - 12.2 \end{align}

(c)

\begin{align} {1 \over x} - 5x & = 5 \\ {1 \over x} & = 5 + 5x \\ 1 & = x(5 + 5x) \\ 1 & = 5x + 5x^2 \\ 0 & = 5x^2 + 5x - 1 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-5 \pm \sqrt{(5)^2 - 4(5)(-1)} \over 2(5)} \\ & = {-5 \pm \sqrt{45} \over 10} \\ & \approx 0.170820 \text{ or } -1.170820 \\ & \approx 0.171 \text{ or } -1.17 \end{align}

(d)

\begin{align} {5 \over x} & = 3 - {x \over x - 3} \\ {5 \over x} & = {3 \over 1} - {x \over x - 3} \\ {5 \over x} & = {3(x - 3) \over x - 3} - {x \over x - 3} \\ {5 \over x} & = {3(x - 3) - x \over x - 3} \\ {5 \over x} & = {3x - 9 - x \over x - 3} \\ {5 \over x} & = {2x - 9 \over x - 3} \\ 5(x - 3) & = x(2x - 9) \\ 5x - 15 & = 2x^2 - 9x \\ 0 & = 2x^2 - 9x - 5x + 15 \\ 0 & = 2x^2 - 14x + 15 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-14) \pm \sqrt{(-14)^2 - 4(2)(15)} \over 2(2)} \\ & = {14 \pm \sqrt{ 76 } \over 4 } \\ & \approx 5.679449 \text{ or } 1.320551 \\ & \approx 5.68 \text{ or } 1.32 \end{align}

(e)

\begin{align} {2 \over x + 1} + {1 \over x - 3} & = 5 \\ {2(x - 3) \over (x + 1)(x - 3)} + {x + 1 \over (x + 1)(x - 3)} & = 5 \\ {2(x - 3) + x + 1 \over (x + 1)(x - 3)} & = 5 \\ {2x - 6 + x + 1 \over x^2 - 3x + x - 3 } & = 5 \\ {3x - 5 \over x^2 - 2x - 3} & = {5 \over 1} \\ 3x - 5 & = 5(x^2 - 2x - 3) \\ 3x - 5 & = 5x^2 - 10x - 15 \\ 0 & = 5x^2 - 10x - 3x - 15 + 5 \\ 0 & = 5x^2 - 13x - 10 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-13) \pm \sqrt{(-13)^2 - 4(5)(-10)} \over 2(5)} \\ & = {13 \pm \sqrt{369} \over 10} \\ & \approx 3.220937 \text{ or } -0.620937 \\ & \approx 3.22 \text{ or } -0.621 \end{align}

(f)

\begin{align} {x \over x + 1} + {1 \over 5} & = {3 \over x - 2} \\ {5x \over 5(x + 1)} + {x + 1 \over 5(x + 1)} & = {3 \over x - 2} \\ {5x + x + 1 \over 5(x + 1)} & = {3 \over x - 2} \\ {6x + 1 \over 5x + 5} & = {3 \over x - 2} \\ (x - 2)(6x + 1) & = 3(5x + 5) \\ 6x^2 + x - 12x - 2 & = 15x + 15 \\ 6x^2 - 11x - 2 & = 15x + 15 \\ 6x^2 - 11x - 15x - 2 - 15 & = 0 \\ 6x^2 - 26x - 17 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-26) \pm \sqrt{(-26)^2 - 4(6)(-17)} \over 2(6)} \\ & = {26 \pm \sqrt{1084} \over 12} \\ & \approx 4.910346 \text{ or } -0.577013 \\ & \approx 4.91 \text{ or } -0.577 \end{align}

(g)

\begin{align} {5 \over x - 2} - {3 \over x^2 - 4} & = {2 \over 7} \\ {5 \over x - 2} - {3 \over \underbrace{(x + 2)(x - 2)}_{a^2 - b^2 = (a + b)(a - b)}} & = {2 \over 7} \\ {5(x + 2) \over (x + 2)(x - 2)} - {3 \over (x + 2)(x - 2)} & = {2 \over 7} \\ {5(x + 2) - 3 \over (x + 2)(x - 2)} & = {2 \over 7} \\ {5x + 10 - 3 \over x^2 - 4} & = {2 \over 7} \\ {5x + 7 \over x^2 - 4} & = {2 \over 7} \\ 7(5x + 7) & = 2(x^2 - 4) \\ 35x + 49 & = 2x^2 - 8 \\ 0 & = 2x^2 - 35x - 8 - 49 \\ 0 & = 2x^2 - 35x - 57 \\ 0 & = (x - 19)(2x + 3) \end{align} \begin{align} x - 19 & = 0 && \text{ or } & 2x + 3 & = 0 \\ x & = 19 &&& 2x & = -3 \\ & &&& x & = -{3 \over 2} \end{align}

(h)

\begin{align} {1 \over 2x + 1} + {x + 3 \over 2x^2 - 5x - 3} & = 2 \\ {1 \over 2x + 1} + {x + 3 \over (2x + 1)(x - 3)} & = 2 \\ {x - 3 \over (2x + 1)(x - 3)} + {x + 3 \over (2x + 1)(x - 3)} & = 2 \\ {x - 3 + x + 3 \over (2x + 1)(x - 3)} & = 2 \\ {2x \over 2x^2 - 5x - 3} & = {2 \over 1} \\ 2x & = 2(2x^2 - 5x - 3) \\ 2x & = 4x^2 - 10x - 6 \\ 0 & = 4x^2 - 10x - 2x - 6 \\ 0 & = 4x^2 - 12x - 6 \\ 0 & = 2x^2 - 6x - 3 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-6) \pm \sqrt{(-6)^2 - 4(2)(-3)} \over 2(2)} \\ & = {6 \pm \sqrt{60} \over 4} \\ & \approx 3.436492 \text{ or } -0.436492 \\ & \approx 3.44 \text{ or } -0.436 \end{align}

(i)

\begin{align} y & = x^2 - 7x + 12 \\ & = (x - 3)(x - 4) \end{align}

(ii)

\begin{align} y & = x^2 - 7x + 12 \phantom{000000} [\text{Minimum curve } \cup] \\ \\ \text{Let } & x = 0, \\ y & = (0)^2 - 7(0) + 12 \\ y & = 12 \phantom{000000000000000.} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = x^2 - 7x + 12 \\ 0 & = (x - 3)(x - 4) \\ \\ x & = 3 \phantom{0} \text{ or } \phantom{0} x = 4 \phantom{0000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {3 + 4 \over 2} \\ x & = 3.5 \\ \\ \text{Let } & x = 3.5, \\ y & = (3.5)^2 - 7(3.5) + 12 \\ y & = -0.25 \\ \\ \text{Minimum point: } & (3.5, -0.25) \end{align}

(i)

\begin{align} y & = - x^2 + 5x - 4 \\ & = - (x^2 - 5x) - 4 \\ & = - \left[ x^2 - 5x + \left(5 \over 2\right)^2 - \left(5 \over 2\right)^2 \right] - 4 \phantom{000000} [\text{Complete the square}] \\ & = - (x^2 - 5x + 2.5^2 - 2.5^2) - 4 \\ & = - [(x - 2.5)^2 - 6.25] - 4 \\ & = - (x - 2.5)^2 + 6.25 - 4 \\ & = - (x - 2.5)^2 + 2.25 \end{align}

(ii)

\begin{align} y & = - x^2 + 5x - 4 \phantom{000000000} [\text{Maximum curve } \cap] \\ y & = - (x - 2.5)^2 + 2.25 \\ \\ \text{Maximum point: } & (2.5, 2.25) \\ \\ \text{Let } & x = 0, \\ y & = - (0 - 2.5)^2 + 2.25 \\ y & = -4 \phantom{000000000000000000} [y \text{-intercept}] \end{align}

(i)

\begin{align} \text{In November, US\$} x & \rightarrow \text{£} 100 \\ \text{US\$} 1 & \rightarrow \text{£} {100 \over x} \\ \text{US\$} 650 & \rightarrow \text{£} {65 \phantom{.} 000 \over x} \\ \\ \\ \text{In December, US\$} (x - 5) & \rightarrow \text{£} 100 \\ \text{US\$} 1 & \rightarrow \text{£} {100 \over x - 5} \\ \text{US\$} 650 & \rightarrow \text{£} {65 \phantom{.} 000 \over x - 5} \\ \\ \\ {65 \phantom{.} 000 \over x - 5} - {65 \phantom{.} 000 \over x} & = 20 \end{align}

(ii)

\begin{align} {65 \phantom{.} 000 \over x - 5} - {65 \phantom{.} 000 \over x} & = 20 \\ {65 \phantom{.} 000 x \over x(x - 5)} - {65 \phantom{.} 000 (x - 5) \over x(x - 5)} & = 20 \\ {65 \phantom{.} 000x - 65 \phantom{.} 000(x - 5) \over x(x - 5)} & = 20 \\ {65 \phantom{.} 000x - 65 \phantom{.} 000x + 325 \phantom{.} 000 \over x^2 - 5x} & = 20 \\ {325 \phantom{.} 000 \over x^2 - 5x} & = {20 \over 1} \\ 325 \phantom{.} 000 & = 20(x^2 - 5x) \\ {325 \phantom{.} 000 \over 20} & = x^2 - 5x \\ 16 \phantom{.} 250 & = x^2 - 5x \\ 0 & = x^2 - 5x - 16 \phantom{.} 250 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-5) \pm \sqrt{(-5)^2 - 4(1)(-16 \phantom{.} 250)} \over 2(1)} \\ & = {5 \pm \sqrt{65 \phantom{.} 025} \over 2} \\ & = 130 \text{ or } -125 \\ \\ \\ \text{In November, £} 100 & \rightarrow \text{US\$} x \\ \text{£} 1250 & \rightarrow \text{US\$} 12.5x \\ \\ \text{Amount of dollars received} & = 12.5x \\ & = 12.5 (130) \\ & = \text{US\$} 1625 \end{align}

(i)

\begin{align} \text{Time} & = { \text{Distance} \over \text{Speed} } \\ \\ \text{Time taken by Ali} & = {40 \over x} \text{ hours} \\ \\ \text{Time taken by Kumar} & = {40 \over x - 30} \text{ hours} \end{align}

(ii)

\begin{align} 10 \text{ mins} & = {1 \over 6} \text{ hours} \\ \\ {40 \over x} + {1 \over 6} & = {40 \over x - 30} \\ {6(40) \over 6x} + {x \over 6x} & = {40 \over x - 30} \\ {240 + x \over 6x} & = {40 \over x - 30} \\ (x - 30)(240 + x) & = 40(6x) \\ 240x + x^2 - 7200 - 30x & = 240x \\ x^2 + 210x - 7200 & = 240x \\ x^2 + 210x - 240x - 7200 & = 0 \\ x^2 - 30x - 7200 & = 0 \phantom{00} \text{ (Shown)} \end{align}

(iii)

\begin{align} x^2 & - 30x - 7200 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-30) \pm \sqrt{(-30)^2 - 4(1)(-7200)} \over 2(1)} \\ & = {30 \pm \sqrt{29 \phantom{.} 700} \over 2} \\ & \approx 101.168440 \text{ or } -71.168440 \\ & \approx 101.17 \text{ or } -71.17 \text{ (2 d.p.)} \end{align}

(iv)

\begin{align} \text{Since Ali's average speed is positive, reject the negative root.} \end{align}

(v)

\begin{align} \text{Time taken by Kumar} & = {40 \over x - 30} \\ & = {40 \over {30 + \sqrt{29700} \over 2} - 30} \\ & \approx 0.562047 \text{ hours} \\ & \approx 33.722813 \text{ mins} \\ & \approx 33.7 \text{ mins} \end{align}

(i)

\begin{align} x \text{ g of sugar} & \rightarrow 1 \text{ cake} \\ 1 \text{ g of sugar} & \rightarrow {1 \over x} \text{ cake} \\ 1 \text{ kg of sugar} & \rightarrow {1000 \over x} \text{ cake} \phantom{000000} [\text{1 kg = 1000 g}] \\ 6 \text{ kg of sugar} & \rightarrow {6000 \over x} \text{ cakes} \end{align}

(ii)

\begin{align} (x + 10) \text{ g of sugar} & \rightarrow 1 \text{ cake} \\ 1 \text{ g of sugar} & \rightarrow {1 \over x + 10} \text{ cake} \\ 1 \text{ kg of sugar} & \rightarrow {1000 \over x + 10} \text{ cake} \\ 6 \text{ kg of sugar} & \rightarrow {6000 \over x + 10} \text{ cakes} \end{align}

(iii)

\begin{align} {6000 \over x} - {6000 \over x + 10} & = 3 \\ {6000(x + 10) \over x(x + 10)} - {6000x \over x(x + 10)} & = 3 \\ {6000(x + 10) - 6000x \over x(x + 10)} & = 3 \\ {6000x + 60 \phantom{.} 000 - 6000x \over x^2 + 10x} & = 3 \\ {60 \phantom{.} 000 \over x^2 + 10x} & = {3 \over 1} \\ 60 \phantom{.} 000 & = 3(x^2 + 10x) \\ {60 \phantom{.} 000 \over 3} & = x^2 + 10x \\ 20 \phantom{.} 000 & = x^2 + 10x \\ 0 & = x^2 + 10x - 20 \phantom{.} 000 \phantom{00} \text{ (Shown)} \end{align}

(iv)

\begin{align} 0 & = x^2 + 10x - 20 \phantom{.} 000 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-10 \pm \sqrt{(10)^2 - 4(1)(-20 \phantom{.} 000)} \over 2(1)} \\ & = {-10 \pm \sqrt{80 \phantom{.} 100} \over 2} \\ & \approx 136.509717 \text{ or } -146.509717 \\ & \approx 137 \text{ or } -147 \end{align}

(v)

\begin{align} \text{Use the positive root, } x & = {-10 + \sqrt{80100} \over 2} \\ \\ {6000 \over x} & \approx 43.952915 \\ \text{Whole cakes baked by Devi} & = 43 \phantom{0000} [\text{Round down}] \\ \\ {6000 \over x + 10} & \approx 40.952915 \\ \text{Whole cakes baked by Siti} & = 40 \phantom{0000} [\text{Round down}] \\ \\ \text{Money collected} & = (43 + 40) \times 15 \\ & = \$ 1245 \end{align}

(i)

\begin{align} \text{Length of floor} & = (35 - 2x) \text{ m} \\ \\ \text{Breadth of floor} & = (22 - 2x) \text{ m} \end{align}

(ii)

\begin{align} \text{Floor area} & = (35 - 2x)(22 - 2x) \\ 400 & = 770 - 70x - 44x + 4x^2 \\ 400 & = 4x^2 - 114x + 770 \\ 0 & = 4x^2 - 114x + 770 - 400 \\ 0 & = 4x^2 - 114x + 370 \\ 0 & = 2x^2 - 57x + 185 \phantom{00} \text{ (Shown)} \end{align}

(iii)

\begin{align} 0 & = 2x^2 - 57x + 185 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-57) \pm \sqrt{(-57)^2 - 4(2)(185)} \over 2(2)} \\ & = {57 \pm \sqrt{1769} \over 4} \\ & \approx 24.764870 \text{ or } 3.735130 \\ & \approx 24.76 \text{ or } 3.74 \text{ (2 d.p.)} \end{align}

(iv) Reject the larger root, since it would make the dimensions of the tiled part negative.

\begin{align} \text{Width of the untiled border} & = x \\ & \approx 3.74 \text{ m} \end{align}

(i)

\begin{align} 60 + 25x - x^2 & = 0 \\ - x^2 + 25x + 60 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-25 \pm \sqrt{(25)^2 - 4(-1)(60)} \over 2(-1)} \\ & = {-25 \pm \sqrt{865} \over -2} \\ & \approx -2.205441 \text{ or } 27.205441 \\ & \approx -2.2 \text{ or } 27.2 \text{ (1 d.p.)} \\ \\ \\ x \approx 27.2 \text{ represents } & \text{the horizontal distance from the tower} \\ \text{when the stone is at } & \text{sea level } (y = 0) \end{align}

(ii)

(iii)(a)

\begin{align} \text{Greatest height } (y) & = 216 \text{ m} \end{align}

(iii)(b)

\begin{align} \text{Horizontal distance } (x) & = 6.5 \text{ m} \text{ or } 18.5 \text{ m} \end{align}

\begin{align} \text{Let the time taken by the} & \text{ larger pipe be } x \text{ mins} \\ \\ x \text{ mins} & \rightarrow 1 \text{ tank} \\ 1 \text{ min} & \rightarrow {1 \over x} \text{ tank} \\ 11{1 \over 9} \text{ mins} & \rightarrow {100 \over 9x} \text{ tank} \phantom{000000} \left[ 11{1 \over 9} = {100 \over 9} \right] \\ \\ \\ \text{Time taken by smaller pipe} & = (x + 5) \text{ mins} \\ \\ (x + 5) \text{ mins} & \rightarrow 1 \text{ tank} \\ 1 \text{ min} & \rightarrow {1 \over x + 5} \text{ tank} \\ 11{1 \over 9} \text{ mins} & \rightarrow {100 \over 9(x + 5)} \text{ tank} \\ \\ \\ {100 \over 9x} + {100 \over 9(x + 5)} & = 1 \\ {100(x + 5) \over 9x(x + 5)} + {100x \over 9x(x + 5)} & = 1 \\ {100(x + 5) + 100x \over 9x(x + 5)} & = 1 \\ {100x + 500 + 100x \over 9x^2 + 45x} & = 1 \\ {200x + 500 \over 9x^2 + 45x} & = {1 \over 1} \\ 200x + 500 & = 9x^2 + 45x \\ 0 & = 9x^2 + 45x - 200x - 500 \\ 0 & = 9x^2 - 155x - 500 \\ 0 & = (x - 20)(9x + 25) \end{align} \begin{align} x - 20 & = 0 && \text{ or } & 9x + 25 & = 0 \\ x & = 20 &&& 9x & = -25 \\ & &&& x & = -{25 \over 9} \text{ (Reject, since } x > 0) \end{align}
\begin{align} \text{Time taken by larger pipe} & = x \\ & = 20 \text{ mins} \\ \\ \text{Time taken by smaller pipe} & = x + 5 \\ & = 20 + 5 \\ & = 25 \text{ mins} \end{align}

\begin{align} \text{Time} & = { \text{Distance} \over \text{Speed} } \\ \\ \text{Time taken (against current)} & = {12 \over x - 5} \text{ hours} \\ \\ \text{Time taken (with current)} & = {12 \over x + 5} \text{ hours} \\ \\ 1 \text{ hour } 30 \text{ mins} & = 1{1 \over 2} \text{ hours} \\ \\ {12 \over x + 5} + {12 \over x - 5} & = {3 \over 2} \\ {12(x - 5) \over (x + 5)(x - 5)} + {12(x + 5) \over (x + 5)(x - 5)} & = {3 \over 2} \\ {12(x - 5) + 12(x + 5) \over (x + 5)(x - 5)} & = {3 \over 2} \\ {12x - 60 + 12x + 60 \over \underbrace{x^2 - 5^2}_{(a + b)(a - b) = a^2 - b^2}} & = {3 \over 2} \\ {24x \over x^2 - 25} & = {3 \over 2} \\ 2(24x) & = 3(x^2 - 25) \\ 48x & = 3x^2 - 75 \\ 0 & = 3x^2 - 48x - 75 \\ 0 & = x^2 - 16x - 25 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-16) \pm \sqrt{(-16)^2 - 4(1)(-25)} \over 2(1)} \\ & = {16 \pm \sqrt{356} \over 2} \\ & \approx 17.433981 \text{ or } -1.433981 \text{ (Reject, since } x > 0) \\ \\ \\ \text{Speed of boat in still water} & = x \\ & = {16 + \sqrt{356} \over 2} \\ & \approx 17.433981 \\ & \approx 17.4 \text{ km/h} \end{align}