(a)

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

(b)

\begin{align} 3x^2 - 20x - 7 & = 0 \\ (3x + 1)(x - 7) & = 0 \end{align} \begin{align} 3x + 1 & = 0 && \text{ or } & x - 7 & = 0 \\ 3x & = -1 &&& x & = 7 \\ x & = -{1 \over 3} \end{align}

(c)

\begin{align} 9x^2 - 25 & = 0 \\ (3x)^2 - (5)^2 & = 0 \\ (3x + 5)(3x - 5) & = 0 \phantom{000000} [a^2 - b^2 = (a + b)(a - b)] \end{align} \begin{align} 3x + 5 & = 0 && \text{ or } & 3x - 5 & = 0 \\ 3x & = -5 &&& 3x & = 5 \\ x & = -{5 \over 3} &&& x & = {5 \over 3} \end{align}

(d)

\begin{align} (x + 3)(x - 2) & = 5x(x - 2) \\ x^2 - 2x + 3x - 6 & = 5x^2 - 10x \\ x^2 + x - 6 & = 5x^2 - 10x \\ 0 & = 5x^2 - x^2 - 10x - x + 6 \\ 0 & = 4x^2 - 11x + 6 \\ 0 & = (4x - 3)(x - 2) \end{align} \begin{align} 4x - 3 & = 0 && \text{ or } & x - 2 & = 0 \\ 4x & = 3 &&& x & = 2 \\ x & = {3 \over 4} \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 - 9 + 4 & = 0 \\ (x - 3)^2 - 5 & = 0 \\ (x - 3)^2 & = 5 \\ x - 3 & = \pm \sqrt{5} \end{align} \begin{align} x - 3 & = \sqrt{5} && \text{ or } & x - 3 & = - \sqrt{5} \\ x & = \sqrt{5} + 3 &&& x & = - \sqrt{5} + 3 \\ x & \approx 5.24 &&& x & \approx 0.764 \end{align}

(b)

\begin{align} x^2 + 4x - 3 & = 0 \\ x^2 + 4x + \left(4 \over 2\right)^2 - \left(4 \over 2\right)^2 - 3 & =0 \\ x^2 + 4x + 2^2 - 2^2 - 3 & = 0 \\ (x + 2)^2 - 4 - 3 & = 0 \\ (x + 2)^2 - 7 & = 0 \\ (x + 2)^2 & = 7 \\ x + 2 & = \pm \sqrt{7} \end{align} \begin{align} x + 2 & = \sqrt{7} && \text{ or } & x + 2 & = - \sqrt{7} \\ x & = \sqrt{7} - 2 &&& x & = - \sqrt{7} - 2 \\ x & \approx 0.646 &&& x & \approx -4.65 \end{align}

(c)

\begin{align} x^2 + 15x + 8 & = 0 \\ x^2 + 15x + \left(15 \over 2\right)^2 - \left(15 \over 2\right)^2 + 8 & = 0 \\ x^2 + 15x + 7.5^2 - 7.5^2 + 8 & = 0 \\ (x + 7.5)^2 - 56.25 + 8 & = 0 \\ (x + 7.5)^2 - 48.25 & = 0 \\ (x + 7.5)^2 & = 48.25 \\ x + 7.5 & = \pm \sqrt{48.25} \end{align} \begin{align} x + 7.5 & = \sqrt{48.25} && \text{ or } & x + 7.5 & = - \sqrt{48.25} \\ x & = \sqrt{48.25} - 7.5 &&& x & = - \sqrt{48.25} - 7.5 \\ x & \approx -0.554 &&& x & \approx -14.4 \end{align}

(d)

\begin{align} x^2 - 7x - 2 & = 0 \\ x^2 - 7x + \left(7 \over 2\right)^2 - \left(7 \over 2\right)^2 - 2 & = 0 \\ x^2 - 7x + 3.5^2 - 3.5^2 - 2 & = 0 \\ (x - 3.5)^2 - 12.25 - 2 & =0 \\ (x - 3.5)^2 - 14.25 & = 0 \\ (x - 3.5)^2 & = 14.25 \\ x - 3.5 & = \pm \sqrt{14.25} \end{align} \begin{align} x - 3.5 & = \sqrt{14.25} && \text{ or } & x - 3.5 & = - \sqrt{14.25} \\ x & = \sqrt{14.25} + 3.5 &&& x & = - \sqrt{14.25} + 3.5 \\ x & \approx 7.27 &&& x & \approx -0.275 \end{align}

(a)

\begin{align} x^2 & + 13x - 88 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-13 \pm \sqrt{(13)^2 - 4(1)(-88)} \over 2(1)} \\ & = {-13 \pm \sqrt{521} \over 2} \\ & = 4.9127 \text{ or } -17.912 \\ & \approx 4.91 \text{ or } -17.9 \end{align}

(b)

\begin{align} 3x^2 & + 12x + 1 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-12 \pm \sqrt{(12)^2 - 4(3)(1)} \over 2(3)} \\ & = {-12 \pm \sqrt{132} \over 6} \\ & = -0.085 \phantom{.} 145 \text{ or } -3.9148 \\ & \approx -0.0851 \text{ or } -3.91 \end{align}

(c)

\begin{align} -4x^2 & + 5x - 9 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-5 \pm \sqrt{(5)^2 - 4(-4)(-9)} \over 2(-4)} \\ & = {-5 \pm \sqrt{-119} \over -8} \\ \\ \\ & \text{No roots for } \sqrt{-119} \\ \\ \therefore & \text{ Equation has no solutions} \end{align}

(d)

\begin{align} -5x^2 & + 8x + 2 = 0 \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-8 \pm \sqrt{(8)^2 - 4(-5)(2)} \over 2(-5)} \\ & = {-8 \pm \sqrt{104} \over -10} \\ & = -0.21980 \text{ or } 1.8198 \\ & \approx -0.220 \text{ or } 1.82 \end{align}

(a)

\begin{align} {x + 1 \over 2} & = {8 \over x + 1} \\ {(x + 1)(x + 1) \over 2(x + 1)} & = {8(2) \over 2(x + 1)} \\ \\ (x + 1)(x + 1) & = 8(2) \\ x^2 + x + x + 1 & = 16 \\ x^2 + 2x + 1 & = 16 \\ x^2 + 2x + 1 - 16 & = 0 \\ x^2 + 2x - 15 & = 0 \\ (x + 5)(x - 3) & =0 \end{align} \begin{align} x + 5 & = 0 && \text{ or } & x - 3 & =0 \\ x & = -5 &&& x & = 3 \end{align}

(b)

\begin{align} {10(x - 2) \over 1} + {10 \over x - 2} & = {29 \over 1} \\ {10(x - 2)(x - 2) \over x - 2} + {10 \over x - 2} & = {29(x - 2) \over x - 2} \\ {10(x - 2)(x - 2) + 10 \over x - 2} & = {29(x - 2) \over x - 2} \\ \\ 10(x - 2)(x - 2) + 10 & = 29(x - 2) \\ 10(x^2 - 2x - 2x + 4) + 10 & = 29x - 58 \\ 10(x^2 - 4x + 4) + 10 & = 29x - 58 \\ 10x^2 - 40x + 40 + 10 & = 29x - 58 \\ 10x^2 - 40x + 50 & = 29x - 58 \\ 10x^2 - 40x - 29x + 50 + 58 & = 0 \\ 10x^2 - 69x + 108 & = 0 \\ (5x - 12)(2x - 9) & = 0 \end{align} \begin{align} 5x - 12 & = 0 && \text{ or } & 2x - 9 & = 0 \\ 5x & = 12 &&& 2x & = 9 \\ x & = {12 \over 5} &&& x & = {9 \over 2} \end{align}

(c)

\begin{align} {x \over x + 1} + {x + 1 \over x} & = {3 \over 1} \\ {x^2 \over x(x + 1)} + {(x + 1)(x + 1) \over x(x + 1)} & = {3x(x + 1) \over x(x + 1)} \\ {x^2 + (x + 1)(x + 1) \over x(x + 1)} & = {3x(x + 1) \over x(x + 1)} \\ \\ x^2 + (x + 1)(x + 1) & = 3x(x + 1) \\ x^2 + x^2 + x + x + 1 & = 3x^2 + 3x \\ 2x^2 + 2x + 1 & = 3x^2 + 3x \\ 0 & = 3x^2 - 2x^2 + 3x - 2x - 1 \\ 0 & = x^2 + x - 1 \\ \\ 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} \\ & = 0.61803 \text{ or } -1.618 \\ & \approx 0.618 \text{ or } -1.62 \end{align}

(d)

\begin{align} {3 \over 7x - 2} - {2 \over 7x + 3} & = {3 \over 4} \\ {3(4)(7x + 3) \over 4(7x - 2)(7x + 3)} - {2(4)(7x - 2) \over 4(7x - 2)(7x + 3)} & = {3(7x - 2)(7x + 3) \over 4(7x - 2)(7x + 3)} \\ {12(7x + 3) - 8(7x - 2) \over 4(7x - 2)(7x + 3)} & = {3(7x - 2)(7x + 3) \over 4(7x - 2)(7x + 3)} \\ \\ 12(7x + 3) - 8(7x - 2) & = 3(7x - 2)(7x + 3) \\ 84x + 36 - 56x + 16 & = 3(49x^2 + 21x - 14x - 6) \\ 28x + 52 & = 3(49x^2 + 7x - 6) \\ 28x + 52 & = 147x^2 + 21x - 18 \\ 0 & = 147x^2 + 21x - 28x - 18 - 52 \\ 0 & = 147x^2 - 7x - 70 \\ 0 & = 21x^2 - x - 10 \\ 0 & = (3x + 2)(7x - 5) \end{align} \begin{align} 3x + 2 & = 0 && \text{ or } & 7x - 5 & = 0 \\ 3x & = -2 &&& 7x & = 5 \\ x & = -{2 \over 3} &&& x & = {5 \over 7} \end{align}

(a)

\begin{align} 3x^2 & + 7x + 1 = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-7 \pm \sqrt{(7)^2 - 4(3)(1)} \over 2(3)} \\ & = {-7 \pm \sqrt{37} \over 6} \\ & = -0.15287 \text{ or } -2.1804 \\ & \approx -0.153 \text{ or } -2.18 \end{align}

(b)

\begin{align} (2x - 1)^2 & = (x + 1)^2 \\ 2x - 1 & = \pm \sqrt{(x + 1)^2} \\ 2x - 1 & = \pm (x + 1) \end{align} \begin{align} 2x - 1 & = x + 1 && \text{ or } & 2x - 1 & = -(x + 1) \\ 2x - x & = 1 + 1 &&& 2x - 1 & = -x - 1 \\ x & = 2 &&& 2x + x & = -1 + 1 \\ & &&& 3x & = 0 \\ & &&& x & = 0 \end{align}

(c)

\begin{align} (x + 3)(2x - 9) & = x(x + 2) \\ 2x^2 - 9x + 6x - 27 & = x^2 + 2x \\ 2x^2 - 3x - 27 & = x^2 + 2x \\ 2x^2 - x^2 - 3x - 2x - 27 & = 0 \\ x^2 - 5x - 27 & = 0 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-5) \pm \sqrt{(-5)^2 - 4(1)(-27)} \over 2(1)} \\ & = {5 \pm \sqrt{133} \over 2} \\ & = 8.2662 \text{ or } -3.2662 \\ & \approx 8.27 \text{ or } -3.27 \end{align}

(d)

\begin{align} {1 \over 3x} + {2 \over 3x - 1} + {3 \over 3x + 1} & = {0 \over 1} \\ {(3x - 1)(3x + 1) \over 3x(3x - 1)(3x + 1)} + {2(3x)(3x + 1) \over 3x(3x - 1)(3x + 1)} + {3(3x)(3x - 1) \over 3x(3x - 1)(3x + 1)} & = {0 \over 3x(3x - 1)(3x + 1)} \\ {(3x - 1)(3x + 1) + 6x(3x + 1) + 9x(3x - 1) \over 3x(3x - 1)(3x + 1)} & = {0 \over 3x(3x - 1)(3x + 1)} \\ (3x - 1)(3x + 1) + 6x(3x + 1) + 9x(3x - 1) & = 0 \\ 9x^2 + 3x - 3x - 1 + 18x^2 + 6x + 27x^2 - 9x & = 0 \\ 54x^2 - 3x - 1 & = 0 \\ (9x + 1)(6x - 1) & = 0 \end{align} \begin{align} 9x + 1 & = 0 && \text{ or } & 6x - 1 & = 0 \\ 9x & = -1 &&& 6x & = 1 \\ x & = -{1 \over 9} &&& x & = {1 \over 6} \end{align}

(a)

\begin{align} x^2 + 8x - 5 & = x^2 + 8x + \left(8 \over 2\right)^2 - \left(8 \over 2\right)^2 - 5 \phantom{000000} [\text{Complete the square}] \\ & = x^2 + 8x + 4^2 - 4^2 - 5 \\ & = (x + 4)^2 - 16 - 5 \\ & = (x + 4)^2 - 21 \end{align}

(b)

\begin{align} x^2 + 8x - 5 & = 0 \\ (x + 4)^2 - 21 & = 0 \\ (x + 4)^2 & = 21 \\ x + 4 & = \pm \sqrt{21} \end{align} \begin{align} x + 4 & = \sqrt{21} && \text{ or } & x + 4 & = - \sqrt{21} \\ x & = \sqrt{21} - 4 &&& x & = - \sqrt{21} - 4 \\ x & \approx 0.583 &&& x & \approx -8.58 \end{align}

(c)

\begin{align} y & = x^2 + 8x - 5 \phantom{000000} [\text{Minimum curve } \cup] \\ y & = (x + 4)^2 - 21 \\ \\ \text{Turning} & \text{ point: } (-4, -21) \\ \\ \text{Let } & x = 0, \\ y & = (0 + 4)^2 - 21 \\ y & = -5 \phantom{0000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = (x + 4)^2 - 21 \\ \\ \text{From (b), } & x \approx 0.583, -8.58 \phantom{0000} [x \text{-intercepts}] \end{align}

(a)

\begin{align} -x^2 + 12x - 35 & = -(x^2 - 12x + 35) \\ & = - (x - 5)(x - 7) \end{align}

(b)

\begin{align} y & = -x^2 + 12x - 35 \phantom{000000} [\text{Maximum curve } \cap] \\ y & = - (x - 5)(x - 7) \\ \\ \text{Let } & x = 0, \\ y & = -(0 - 5)(0 - 7) \\ y & = - 35 \phantom{0000000000000000} [y \text{-intercept}] \\ \\ \text{Let } & y = 0, \\ 0 & = - (x - 5)(x - 7) \\ \\ x - 5 & = 0 \phantom{0} \text{ or } \phantom{0} x - 7 = 0 \\ x & = 5 \phantom{00000000} x = 7 \phantom{000000} [x \text{-intercepts}] \\ \\ \text{Line of symmetry, } x & = {5 + 7 \over 2} \\ x & = 6 \\ \\ \text{Let } & x = 6, \\ y & = -(6 - 5)(6 - 7) \\ y & = 1 \\ \\ \text{Turning} & \text{ point: } (6, 1) \end{align}

(a)

x	-5	-4	-3	-2	-1	0	1	2
y	4	-2	-6	-8	-8	-6	-2	4

(b)

\begin{align} \underbrace{x^2 + 3x - 6}_\text{Curve} & = 0 \\ \\ \text{From graph, } & x = -4.4, 1.4 \phantom{000000} [\text{Look for } x \text{-intercepts}] \end{align}

(c)(i)

\begin{align} x^2 + 3x - 3 & = 0 \\ x^2 + 3x - 3 - 3 & = 0 - 3 \\ \underbrace{x^2 + 3x - 6}_\text{Curve} & = -3 \\ \\ \text{Draw } & y = -3 \\ \\ \text{From graph, } & x = -3.8, 0.8 \end{align}

(c)(iI)

(a)

x	-3	-2	-1	0	1	2
y	-7	1	5	5	1	-7

(b)

\begin{align} \underbrace{5 - 2x - 2x^2}_\text{Curve} & = 0 \\ \\ \text{From graph, } & x = -2.15, 1.15 \phantom{000000} [\text{Look for } x \text{-intercepts}] \end{align}

(c)(i)

\begin{align} 1 - 2x - 2x^2 & = 0 \\ 1 - 2x - 2x^2 + 4 & = 0 + 4 \\ \underbrace{5 - 2x - 2x^2}_\text{Curve} & = 4 \\ \\ \text{Draw } & y = 4 \\ \\ \text{From graph, } & x = -1.35, 0.35 \end{align}

(c)(iI)

(a)

\begin{align} y & = x^2 + 4x \\ \\ \text{Let } & y = 0, \\ 0 & = x^2 + 4x \\ 0 & = x(x + 4) \end{align} \begin{align} x & = 0 && \text{ or } & x + 4 & = 0 \\ & &&& x & = -4 \end{align}
$$ \therefore A(-4, 0) $$

(b)

\begin{align} x & = {-4 + 0 \over 2} \\ x & = -2 \end{align}

(c)

\begin{align} \text{Substitute } & x = - 2 \text{ into } y = x^2 + 4x, \\ y & = (-2)^2 + 4(-2) \\ y & = -4 \\ \\ \text{Minimum} & \text{ point: } (-2, -4) \end{align}

\begin{align} \text{Area of lawn} & = 2x \times x \\ & = 2x^2 \text{ m}^2 \\ \\ \text{Areae of 1 path} & = 1.5 \times x \\ & = 1.5x \text{ m}^2 \\ \\ \text{Total area} & = 2x^2 + 2(1.5)x \\ 189 & = 2x^2 + 3x \\ 0 & = 2x^2 + 3x - 189 \\ 0 & = (x - 9)(2x + 21) \end{align} \begin{align} x - 9 & = 0 && \text{ or } & 2x + 21 & = 0 \\ x & = 9 &&& 2x & = -21 \\ & &&& x & = -{21 \over 2} \text{ (Reject, since } x > 0) \end{align}
\begin{align} \text{Breadth of lawn} & = x \\ & = 9 \text{ m} \\ \\ \text{Length of lawn} & = 2x \\ & = 2(9) \\ & = 18 \text{ m} \end{align}

\begin{align} \text{Area of bed} & = 30 \times 20 \\ & = 600 \text{ m}^2 \\ \\ \text{Area of path} & = {1 \over 3} \times 600 \\ & = 200 \text{ m}^2 \\ \\ \text{Let } x \text{ represent } & \text{the width of the path} \\ \\ \text{Length of bed & path} & = (30 + 2x) \text{ m} \\ \\ \text{Breadth of bed & path} & = (20 + 2x) \text{ m} \\ \\ \text{Area of bed & path} & = (30+2x) (20 + 2x) \\ 600 + 200 & = 600 + 60x + 40x + 4x^2 \\ 800 & = 4x^2 + 100x + 600 \\ 0 & = 4x^2 + 100x + 600 - 800 \\ 0 & = 4x^2 + 100x - 200 \\ 0 & = x^2 + 25x - 50 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-25 \pm \sqrt{(25)^2 - 4(1)(-50)} \over 2(1)} \\ & = {-25 \pm \sqrt{825} \over 2} \\ & = 1.8614 \text{ or } -26.861 \text{ (Reject, since } x > 0) \\ \\ \text{Width of path} & = x \\ & = 1.8614 \\ & \approx 1.86 \text{ m} \end{align}

\begin{align} \text{Let } x \text{ denote the ini} & \text{tial speed (in km/h) of the train} \\ \\ \text{Time} & = {\text{Distance} \over \text{Speed}} \\ \\ \text{Time taken (initial)} & = {180 \over x} \text{ h} \\ \\ \text{Time taken (after)} & = {180 \over x + 10} \text{ h} \\ \\ 15 \text{ mins} & = {15 \over 60} \text{ h} \\ & = {1 \over 4} \text{ h} \\ \\ {180 \over x} - {180 \over x + 10} & = {1 \over 4} \\ {180(4)(x + 10) \over 4x(x + 10)} - {180(4)(x) \over 4x(x + 10)} & = {x(x + 10) \over 4x(x + 10)} \\ {720(x + 10) \over 4x(x + 10)} - {720x \over 4x(x + 10)} & = {x(x + 10) \over 4x(x + 10)} \\ \\ 720(x + 10) - 720x & = x(x + 10) \\ 720x + 7200 - 720x & = x^2 + 10x \\ 7200 & = x^2 + 10x \\ 0 & = x^2 + 10x - 7200 \\ 0 & = (x - 80)(x + 90) \end{align} \begin{align} x - 80 & = 0 && \text{ or } & x + 90 & = 0 \\ x & = 80 &&& x & = -90 \text{ (Reject, since } x > 0) \end{align}
$$ \text{Average speed} = x = 80 \text{ km/h} $$

(a)(i)

\begin{align} \text{Time} & = {\text{Distance} \over \text{Speed}} \\ \\ \text{Time taken (Abel)} & = {42 \over x} \text{ h} \end{align}

(a)(ii)

\begin{align} \text{Time taken (Sam)} & = {42 \over x + 3} \text{ h} \end{align}

(b)

\begin{align} 28 \text{ mins} & = {28 \over 60} \text{ h} \\ & = {7 \over 15} \text{ h} \\ \\ {42 \over x} - {42 \over x + 3} & = {7 \over 15} \\ {42(15)(x + 3) \over 15x(x + 3)} - {42(15)(x) \over 15x(x + 3)} & = {7x(x + 3) \over 15x(x + 3)} \\ {630(x + 3) - 630x \over 15x(x + 3)} & = {7x(x + 3) \over 15x(x + 3)} \\ \\ 630(x + 3) - 630x & = 7x(x + 3) \\ 630x + 1890 - 630x & = 7x^2 + 21x \\ 1890 & = 7x^2 + 21x \\ 0 & = 7x^2 + 21x - 1890 \\ 0 & = x^2 + 3x - 270 \phantom{00} \text{ (Shown)} \end{align}

(c)

\begin{align} 0 & = x^2 + 3x - 270 \\ 0 & = (x - 15)(x + 18) \end{align} \begin{align} x - 15 & = 0 && \text{ or } & x + 18 & = 0 \\ x & = 15 &&& x & = -18 \end{align}

(d) Use x = 15, since x (Abel's speed) cannot be a negative value

\begin{align} \text{From (a)(i), time taken (Abel)} & = {42 \over x} \text{ h} \\ & = {42 \over 15} \text{ h} \\ & = 2.8 \text{ h} \\ & = 2 \text{ h } (0.8 \times 60) \text{ mins} \\ & = 2 \text{ h } 48 \text{ mins} \end{align}

(a)(i)

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

(a)(ii)

\begin{align} (2x - 1) \text{ mins} & \rightarrow 135 \text{ doughnuts} \\ 1 \text{ min} & \rightarrow {135 \over 2x - 1} \text{ doughnuts} \end{align}

(b)

\begin{align} {60 \over x} + {135 \over 2x - 1} & = 27 \\ {60 \over x} + {135 \over 2x - 1} & = {27 \over 1} \\ {60(2x - 1) \over x(2x - 1)} + {135x \over x(2x - 1)} & = {27x(2x - 1) \over x(2x - 1)} \\ {60(2x - 1) + 135x \over x(2x - 1)} & = {27x(2x - 1) \over x(2x - 1)} \\ \\ 60(2x - 1) + 135x & = 27x(2x - 1) \\ 120x - 60 + 135x & = 54x^2 - 27x \\ 255x - 60 & = 54x^2 - 27x \\ 0 & = 54x^2 - 27x - 255x + 60 \\ 0 & = 54x^2 - 282x + 60 \\ 0 & = 9x^2 - 47x + 10 \phantom{00} \text{ (Shown)} \end{align}

(c)

\begin{align} 0 & = 9x^2 - 47x + 10 \\ 0 & = (x - 5)(9x - 2) \end{align} \begin{align} x - 5 & = 0 && \text{ or } & 9x - 2 & = 0 \\ x & = 5 &&& 9x & = 2 \\ & &&& x & = {2 \over 9} \end{align}

(d) Use x = 5 instead of x = ²/₉ - if x = ²/₉, Machine B will take -⁵/₉ minutes to make 135 doughnuts

\begin{align} \text{From (a)(i), } 1 \text{ min} \rightarrow {60 \over x} & = {60 \over 5} = 12 \text{ doughnuts} \\ \\ \text{Time taken to make 90 doughnuts} & = {90 \over 12} \\ & = 7.5 \text{ mins} \end{align}

(a)

$$ {1200 \over x} \text{ mins} $$

(b)

$$ {1200 \over x + 10} \text{ mins} $$

(c)

\begin{align} {1200 \over x} - {1200 \over x + 10} & = 4 \\ {1200 \over x} - {1200 \over x + 10} & = {4 \over 1} \\ {1200(x + 10) \over x(x + 10)} - {1200x \over x(x + 10)} & = {4x(x + 10) \over x(x + 10)} \\ {1200(x + 10) - 1200x \over x(x + 10)} & = {4x(x + 10) \over x(x + 10)} \\ \\ 1200(x + 10) - 1200x & = 4x(x + 10) \\ 1200x + 12 \phantom{.} 000 - 1200x & = 4x^2 + 40x \\ 12 \phantom{.} 000 & = 4x^2 + 40x \\ 0 & = 4x^2 + 40x - 12 \phantom{.} 000 \\ 0 & = x^2 + 10x - 3000 \phantom{00} \text{ (Shown)} \end{align}

(d)

\begin{align} 0 & = x^2 + 10x - 3000 \\ 0 & = (x - 50)(x + 60) \end{align} \begin{align} x - 50 & = 0 && \text{ or } & x + 60 & = 0 \\ x & = 50 &&& x & = -60 \end{align}

(e) Use x = 50 instead of x = -60 since rate of water delivery cannot be negative

\begin{align} \text{From (b), time taken} & = {1200 \over x + 10} \\ & = {1200 \over 50 + 10} \\ & = 20 \text{ mins} \end{align}

(a)

x	0	10	20	30	40	50
y	-225	175	375	375	175	-225

(b)(i)

\begin{align} \text{Max. point: } & (25, 400) \\ \\ \text{Max. profit } (y) & = \$ 400 \\ \\ \text{Selling price } (x) & = \$ 25 \end{align}

(b)(ii) No profit or loss means y = 0

\begin{align} \text{From graph, selling prices} & = \$ 5 \text{ or } \$ 45 \end{align}

(b)(iii)

\begin{align} \text{From graph, selling prices} & = \$ 20 \text{ or } \$ 30 \end{align}

(a)(i)

\begin{align} \text{Radius of large circle} & = {12 \over 2} \\ & = 6 \text{ cm} \\ \\ \text{Radius of small circle} & = x \text{ cm} \\ \\ PQ & = (6 + x) \text{ cm} \end{align}

(a)(ii)

\begin{align} PS & = 6 \text{ cm} \\ \\ TS & = QR = x \text{ cm} \\ \\ PT & = (6 - x) \text { cm} \end{align}

(a)(iii)

\begin{align} DS & = \text{Radius of large circle} \\ & = 6 \text{ cm} \\ \\ RC & = \text{Radius of small circle} \\ & = x \text{ cm} \\ \\ SR & = 18 - 6 - x \\ & = (12 - x) \text{ cm} \end{align}

(b)

\begin{align} \text{By Pytha} & \text{goras theorem, } \\ PQ^2 & = PT^2 + TQ^2 \\ (6 + x)^2 & = (6 - x)^2 + (12 - x)^2 \\ (6 + x)(6 + x) & = (6 - x)(6 - x) + (12 - x)(12 - x) \\ 36 + 6x + 6x + x^2 & = 36 - 6x - 6x + x^2 + 144 - 12x - 12x + x^2 \\ x^2 + 12x + 36 & = 2x^2 - 36x + 180 \\ 0 & = 2x^2 - x^2 - 36x - 12x + 180 - 36 \\ 0 & = x^2 - 48x + 144 \phantom{00} \text{ (Shown)} \end{align}

(c)

\begin{align} 0 & = x^2 - 48x + 144 \\ \\ x & = {-b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-48) \pm \sqrt{(-48)^2 - 4(1)(144)} \over 2(1)} \\ & = {48 \pm \sqrt{1728} \over 2} \\ & = 44.7846 \text{ or } 3.2153 \\ \\ [\text{Do } & \text{not use } x = 44.7846 \text{ since it is longer than the sides of the rectangle}] \\ \\ \text{Radius of small circle} & = x \\ & = 3.2153 \\ & \approx 3.22 \text{ cm} \end{align}