(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 & = 4.1583 &&& y & = 0.84168 \\ 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 & = 0.60555 &&& x & = -6.6055 \\ 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 & = -0.80741 &&& y & = -6.1925 \\ y & \approx -0.81 \text{ (2 d.p.)} &&& x & \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 & = 1.6180 &&& z & = -0.61803 \\ z & \approx 1.62 \text{ (2 d.p.)} &&& x & \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 & = 4.7201 &&& x & = -5.7201 \\ 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} \\ & = 1.2655 \text{ or } - 2.76555 \\ & \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} \\ & = -0.11651 \text{ or } 1.7165 \\ & \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} \\ & = 1.4683 \text{ or } -1.1350 \\ & \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} \\ & = -7.1400 \text{ or } 0.140 \phantom{.} 054 \\ & \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} \\ & = 4.7320 \text{ or } 1.2679 \\ & \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} \\ & = 5.2360 \text{ or } 0.76393 \\ & \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} \\ & = 2.3507 \text{ or } -0.85078 \\ & \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 \\ \\ \\ p & = {-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} \\ & = 4.3122 \text{ or } 2.5766 \\ & \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} \\ & = 0.41968 \text{ or } 11.913 \\ & \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} \\ & = -5.7904 \text{ or } 2.5904 \\ & \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} \\ & = 3.9128 \text{ or } 2.0871 \\ & \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} \\ & = 0.21922 \text{ or } 2.2807 \\ & \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} \\ & = -0.36980 \text{ or } -3.3801 \\ & \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} \\ & = 7.9278 \text{ or } 1.0721 \\ & \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 = 1.0721 \text{ is the only possible solution} \end{align}

(v)

\begin{align} \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 - 1.0721) \times 1.0721 \times 7 \\ & = 29.748 \\ & \approx 29.7 \text{ m}^3 \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} \\ & = 126.162 \text{ or } -133.162 \\ & \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 126.162 + 7} \\ & = 4.5057 \text{ hours} \\ & = 4 \text{ hours } (0.5057 \times 60) \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}