(a)

\begin{align} x^2 & - 12x + 36 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-12) \pm \sqrt{(-12)^2 - 4(1)(36)} \over 2(1)} \\ & = {12 \pm \sqrt{0} \over 2} \\ & = 6 \end{align}

(b)

\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} \\ & = -0.17712 \text{ or } -2.8228 \\ & \approx -0.18 \text{ or } -2.82 \text{ (2 d.p.)} \end{align}

(c)

\begin{align} 3x^2 & - 2x - 5 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-2) \pm \sqrt{(-2)^2 - 4(3)(-5)} \over 2(3)} \\ & = {2 \pm \sqrt{64} \over 6} \\ & = {4 \over 3} \text{ or } -1 \end{align}

(d)

\begin{align} 5x^2 & - x + 4 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-1) \pm \sqrt{ (-1)^2 - 4(5)(4)} \over 2(5)} \\ & = {1 \pm \sqrt{-79} \over 10} \\ \\ \\ \sqrt{-79} & \text{ is not a real number} \\ \\ \therefore \text{Equa} & \text{tion has no solutions} \end{align}

(e)

\begin{align} 7x^2 & + x - 2 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 1 \pm \sqrt{ (1)^2 - 4(7)(-2)} \over 2(7)} \\ & = {-1 \pm \sqrt{57} \over 14} \\ & = 0.46784 \text{ or } -0.6107 \\ & \approx 0.47 \text{ or } - 0.61 \text{ (2 d.p.)} \end{align}

(a)

\begin{align} x + 7x(1 - x) & = 3x - 26 \\ x + 7x - 7x^2 & = 3x - 26 \\ 8x - 7x^2 & = 3x - 26 \\ 0 & = 7x^2 + 3x - 8x - 26 \\ 0 & = 7x^2 - 5x - 26 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-5) \pm \sqrt{ (-5)^2 - 4(7)(-26)} \over 2(7)} \\ & = {5 \pm \sqrt{753} \over 14} \\ & = 2.3172 \text{ or } -1.6029 \\ & \approx 2.32 \text{ or } -1.60 \end{align}

(b)

\begin{align} (4x - 5)(x + 2) & = 4x + 1 \\ 4x^2 + 8x - 5x - 10 & = 4x + 1 \\ 4x^2 + 3x - 10 & = 4x + 1 \\ 4x^2 + 3x - 4x - 10 - 1 & = 0 \\ 4x^2 - x - 11 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-1) \pm \sqrt{(-1)^2 - 4(4)(-11)} \over 2(4)} \\ & = {1 \pm \sqrt{177} \over 8} \\ & = 1.788 \text{ or } -1.538 \\ & \approx 1.79 \text{ or } -1.54 \end{align}

(c)

\begin{align} x(1 - 2x) + 9 & = 0 \\ x - 2x^2 + 9 & = 0 \\ - 2x^2 + x + 9 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- 1 \pm \sqrt{(1)^2 - 4(-2)(9)} \over 2(-2)} \\ & = {-1 \pm \sqrt{73} \over -4} \\ & = -1.886 \text{ or } 2.386 \\ & \approx -1.89 \text{ or } 2.39 \end{align}

(d)

\begin{align} (1 - x)(4 + x) & = 5 \\ 4 + x - 4x - x^2 & = 5 \\ 4 - 3x - x^2 & = 5 \\ 0 & = x^2 + 3x + 5 - 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} \\ & = -0.38196 \text{ or } -2.618 \\ & \approx -0.382 \text{ or } -2.62 \end{align}

(e)

\begin{align} x^2 & = {3 \over 4}x + 2 \\ x^2 - {3 \over 4}x - 2 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- \left(-{3 \over 4}\right) \pm \sqrt{ \left(-{3 \over 4}\right)^2 - 4(1)(-2)} \over 2(1)} \\ & = { {3 \over 4} \pm \sqrt{ 137 \over 16} \over 2 } \\ & = 1.838 \text{ or } -1.088 \\ & \approx 1.84 \text{ or } -1.09 \end{align}

(a)

\begin{align} (x - 2)(x + 1) & = 8 \\ x^2 + x - 2x - 2 & = 8 \\ x^2 - x - 2 & = 8 \\ x^2 - x - 2 - 8 & = 0 \\ x^2 - x - 10 & = 0 \\ \\ 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} \\ & = 3.7015 \text{ or } -2.7015 \\ & \approx 3.70 \text{ or } -2.70 \text{ (2 d.p.)} \end{align}

(b)

\begin{align} 3x^2 - 4 & = 7x \\ 3x^2 - 7x - 4 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-7) \pm \sqrt{ (-7)^2 - 4(3)(-4)} \over 2(3)} \\ & = {7 \pm \sqrt{97} \over 6} \\ & = 2.8081 \text{ or } -0.4748 \\ & \approx 2.81 \text{ or } -0.47 \text{ (2 d.p.)} \end{align}

(c)

\begin{align} (x - 2)(x + 2) & = 4(x - 2) \\ x^2 - 2^2 & = 4x - 8 \phantom{000000} [ (a - b)(a + b) = a^2 - b^2] \\ x^2 - 4 & = 4x - 8 \\ x^2 - 4x - 4 + 8 & = 0 \\ x^2 - 4x + 4 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-4) \pm \sqrt{ (-4)^2 - 4(1)(4)} \over 2(1)} \\ & = {4 \pm \sqrt{0} \over 2} \\ & = 2 \end{align}

(d)

\begin{align} 5x(x + 1) & = 3(x - 2) \\ 5x^2 + 5x & = 3x - 6 \\ 5x^2 + 5x - 3x + 6 & = 0 \\ 5x^2 + 2x + 6 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-2 \pm \sqrt{(2)^2 - 4(5)(6)} \over 2(5)} \\ & = {-2 \pm \sqrt{-116} \over 10} \\ \\ \\ \sqrt{-116} \text{ is } & \text{not a real number} \\ \\ \therefore \text{Equation } & \text{has no solutions} \end{align}

(e)

\begin{align} (2x + 1)^2 & = (x - 3)^2 \\ \underbrace{(2x)^2 + 2(2x)(1) + (1)^2}_{(a + b)^2 = a^2 + 2ab + b^2} & = \underbrace{(x)^2 - 2(x)(3) + (3)^2}_{(a - b)^2 = a^2 - 2ab + b^2} \\ 4x^2 + 4x + 1 & = x^2 - 6x + 9 \\ 4x^2 - x^2 + 4x + 6x + 1 - 9 & = 0 \\ 3x^2 + 10x - 8 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-10 \pm \sqrt{(10)^2 - 4(3)(-8)} \over 2(3)} \\ & = {-10 \pm \sqrt{196} \over 6} \\ & = {2 \over 3} \text{ or } -4 \end{align}

(f)

\begin{align} x(x - 6) & = {2 \over 3}(x^2 + 1) \\ x^2 - 6x & = {2 \over 3}x^2 + {2 \over 3} \\ x^2 - {2 \over 3}x^2 - 6x - {2 \over 3} & = 0 \\ {1 \over 3}x^2 - 6x - {2 \over 3} & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-6) \pm \sqrt{ (-6)^2 - 4 \left(1 \over 3\right) \left(-{2 \over 3}\right)} \over 2 \left(1 \over 3\right)} \\ & = {6 \pm \sqrt{ 332 \over 9} \over {2 \over 3}} \\ & = 18.110 \text{ or } -0.1104 \\ & \approx 18.11 \text{ or } -0.11 \text{ (2 d.p.)} \end{align}

(a)

\begin{align} {1 \over 2}x^2 & + 3x + q = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-3 \pm \sqrt{(3)^2 - 4\left(1 \over 2\right)(q)} \over 2 \left(1 \over 2\right) } \\ & = {-3 \pm \sqrt{9 - 2q} \over 1} \\ & = {-3 \pm \sqrt{9 - 2q} } \\ \\ \\ \implies p + \sqrt{7} & = -3 + \sqrt{9 - 2q} \end{align} \begin{align} p & = -3 &&& 7 & = 9 - 2q \\ & &&& 2q & = 9 - 7 \\ & &&& 2q & = 2 \\ & &&& q & = 1 \end{align}

(b)

\begin{align} x & = -3 - \sqrt{9 - 2q} \\ & = -3 - \sqrt{9 - 2(1)} \\ & = -3 - \sqrt{7} \\ & \approx -5.65 \end{align}

Method 1: Quadratic formula

\begin{align} x^2 & - 6x + 14 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-6) \pm \sqrt{ (-6)^2 - 4(1)(14)} \over 2(1)} \\ & = {6 \pm \sqrt{-20} \over 2} \\ \\ \\ \sqrt{-20} &\text{ is } \text{not a real number} \\ \\ \therefore \text{Equa} & \text{tion has no solutions} \end{align}

Method 2: Complete the square

\begin{align} x^2 - 6x + 14 & = 0 \\ x^2 - 6x + \left(6 \over 2\right)^2 - \left(6 \over 2\right)^2 + 14 & = 0 \\ x^2 - 6x + 3^2 - 3^2 + 14 & = 0 \\ (x - 3)^2 - 9 + 14 & = 0 \\ (x - 3)^2 + 5 & = 0 \\ (x - 3)^2 & = -5 \\ x - 3 & = \pm \sqrt{-5} \\ \\ \\ \sqrt{-5} \text{ is } & \text{not a real number} \\ \\ \therefore \text{Equation} & \text{ has no solutions} \end{align}