(a)

\begin{align} 3x^2 + 10x + 6 & = 0 \\ x^2 + {10 \over 3}x + 2 & = 0 \phantom{0000000} [\text{Divide by 3}] \\ \left(x + {5 \over 3}\right)^2 - \left(5 \over 3\right)^2 + 2 & = 0 \\ \left(x + {5 \over 3}\right)^2 - {7 \over 9} & = 0 \\ \left(x + {5 \over 3}\right)^2 & = {7 \over 9} \\ x + {5 \over 3} & = \pm \sqrt{7 \over 9} \\ x & = \sqrt{7 \over 9} - {5 \over 3} \text{ or } - \sqrt{7 \over 9} - {5 \over 3} \\ x & \approx -0.78 \text{ or } - 2.55 \end{align}

(b)

\begin{align} 4x^2 - 17x + 12 & = 0 \\ x^2 - {17 \over 4}x + 3 & = 0 \phantom{0000000} [\text{Divide by 4}] \\ \left(x - {17 \over 8}\right)^2 - \left(17 \over 8\right)^2 + 3 & = 0 \\ \left(x - {17 \over 8}\right)^2 - {97 \over 64} & = 0 \\ \left(x - {17 \over 8}\right)^2 & = {97 \over 64} \\ x - {17 \over 8} & = \pm \sqrt{97 \over 64} \\ x & = \sqrt{97 \over 64} + {17 \over 8} \text{ or } -\sqrt{97 \over 64} + {17 \over 8} \\ x & \approx 3.36 \text{ or } 0.89 \end{align}

(c)

\begin{align} -5x^2 + 12x - 2 & = 0 \\ 5x^2 - 12x + 2 & = 0 \\ x^2 - {12 \over 5}x + {2 \over 5} & = 0 \phantom{0000000} [\text{Divide by 5}] \\ \left(x - {6 \over 5}\right)^2 - \left(6 \over 5\right)^2 + {2 \over 5} & = 0 \\ \left(x - {6 \over 5}\right)^2 - {26 \over 25} & = 0 \\ \left(x - {6 \over 5}\right)^2 & = {26 \over 25} \\ x - {6 \over 5} & = \pm \sqrt{26 \over 25} \\ x & = \sqrt{26 \over 25} + {6 \over 5} \text{ or } -\sqrt{26 \over 25} + {6 \over 8} \\ x & \approx 2.22 \text{ or } 0.18 \end{align}

(d)

\begin{align} (3x - 2)(2x + 9) & = 2 \\ 6x^2 + 27x - 4x - 18 & = 2 \\ 6x^2 + 23x - 20 & = 0 \\ x^2 + {23 \over 6}x - {10 \over 3} & = 0 \phantom{0000000} [\text{Divide by 6}] \\ \left(x + {23 \over 12} \right)^2 - \left(23 \over 12\right)^2 - {10 \over 3} & = 0 \\ \left(x + {23 \over 12} \right)^2 - {1009 \over 144} & = 0 \\ \left(x + {23 \over 12} \right)^2 & = {1009 \over 144} \\ x + {23 \over 12} & = \pm \sqrt{1009 \over 144} \\ x & = \sqrt{1009 \over 144} - {23 \over 12} \text{ or } -\sqrt{1009 \over 144} - {23 \over 12} \\ x & \approx 0.73 \text{ or } -4.56 \end{align}

(a)

\begin{align} 9x^2 + 8x + 10 & = 0 \\ x^2 + {8 \over 9}x + {10 \over 9} & = 0 \phantom{0000000} [\text{Divide by 9}] \\ \left(x + {4 \over 9}\right)^2 - \left(4 \over 9\right)^2 + {10 \over 9} & = 0 \\ \left(x + {4 \over 9}\right)^2 + {74 \over 81} & = 0 \\ \left(x + {4 \over 9}\right)^2 & = - {74 \over 81} \\ \\ \\ \text{For all real values of } x, & \phantom{.} \left(x + {4 \over 9}\right)^2 \ge 0 \\ \\ \therefore \left(x + {4 \over 9}\right)^2 = - {74 \over 81} & \text{ has no real solutions} \end{align}

(b)

\begin{align} 2x^2 - x + 7 & = 0 \\ x^2 - {1 \over 2}x + {7 \over 2} & = 0 \phantom{0000000} [\text{Divide by 2}] \\ \left(x - {1 \over 4}\right)^2 - \left(1 \over 4\right)^2 + {7 \over 2} & = 0 \\ \left(x - {1 \over 4}\right)^2 + {55 \over 16} & = 0 \\ \left(x - {1 \over 4}\right)^2 & = - {55 \over 16} \\ \\ \\ \text{For all real values of } x, & \phantom{.} \left(x - {1 \over 4}\right)^2 \ge 0 \\ \\ \therefore \left(x - {1 \over 4}\right)^2 = - {55 \over 16} & \text{ has no real solutions} \end{align}

(c)

\begin{align} -6x^2 + 4x - 55 & = 0 \\ 6x^2 - 4x + 55 & = 0 \\ x^2 - {2 \over 3}x + {55 \over 6} & = 0 \phantom{0000000} [\text{Divide by 6}] \\ \left(x - {1 \over 3}\right)^2 - \left(1 \over 3\right)^2 + {55 \over 6} & = 0 \\ \left(x - {1 \over 3}\right)^2 + {163 \over 18} & = 0 \\ \left(x - {1 \over 3}\right)^2 & = - {163 \over 18} \\ \\ \\ \text{For all real values of } x, & \phantom{.} \left(x - {1 \over 3}\right)^2 \ge 0 \\ \\ \therefore \left(x - {1 \over 3}\right)^2 = - {163 \over 18} & \text{ has no real solutions} \end{align}

(d)

\begin{align} 4(5x + 1)^2 & = x - 2020 \\ 4[(5x)^2 + 2(5x)(1) + (1)^2] & = x - 2020 \phantom{0000000} [(a + b)^2 = a^2 + 2ab + b^2] \\ 4(25x^2 + 10x + 1) & = x - 2020 \\ 100x^2 + 40x + 4 & = x - 2020 \\ 100x^2 + 39x + 2024 & = 0 \\ x^2 + 0.39x + 20.24 & = 0 \phantom{00000000000000} [\text{Divide by 100}] \\ \left(x + {0.39 \over 2}\right)^2 - \left(0.39 \over 2\right)^2 + 20.24 & = 0 \\ (x + 0.195)^2 + 20.201 \phantom{.} 975 & = 0 \\ (x + 0.195)^2 & = - 20.201 \phantom{.} 975 \\ \\ \\ \text{For all real values of } x, & \phantom{.} (x + 0.195)^2 \ge 0 \\ \\ \therefore (x + 0.195)^2 = - 20.201 \phantom{.} 975 & \text{ has no real solutions} \end{align}

(a)

\begin{align} -10x^2 &- 15x + 4 = 0 \\ \\ x & = { -b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-15) \pm \sqrt{(-15)^2 - 4(-10)(4)} \over 2(-10)} \\ & = {15 \pm \sqrt{385} \over -20} \\ & \approx -1.73 \text{ or } 0.231 \end{align}

(b)

\begin{align} 5x^2 - & 3x - 86 = 0 \\ \\ x & = { -b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-3) \pm \sqrt{(-3)^2 - 4(5)(-86)} \over 2(5)} \\ & = {3 \pm \sqrt{1729} \over 10} \\ & \approx 4.46 \text{ or } -3.86 \end{align}

(c)

\begin{align} 81 + 64x^2 & = 144x \\ 0 & = -64x^2 + 144x - 81 \\ \\ x & = { -b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(144) \pm \sqrt{(144)^2 - 4(-64)(-81)} \over 2(-64)} \\ & = {-144 \pm \sqrt{0} \over -128} \\ & = {9 \over 8} \end{align}

(d)

\begin{align} 3x^2 + 11 & = 2x (2 - x) \\ 3x^2 + 11 & = 4x - 2x^2 \\ 5x^2 - 4x + 11 & = 0 \\ \\ x & = { -b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {-(-4) \pm \sqrt{(-4)^2 - 4(5)(11)} \over 2(5)} \\ & = {4 \pm \sqrt{-204} \over 10} \\ \\ \text{No real roots} & \text{ since } \sqrt{-204} \text{ is undefined} \end{align}

\begin{align} -x^2 - 4x + 2h & = 7 \\ -x^2 - 4x + 2h - 7 & = 0 \\ \\ b^2 - 4ac & = (-4)^2 - 4(-1)(2h - 7) \\ & = 16 + 4(2h - 7) \\ & = 16 + 8h - 28 \\ & = 8h - 12 \\ \\ b^2 - 4ac & \ge 0 \phantom{000000} [\text{Two real roots - can be distinct or equal}] \\ 8h - 12 & \ge 0 \\ 8h & \ge 12 \\ h & \ge {12 \over 8} \\ h & \ge {3 \over 2} \end{align}

\begin{align} px^2 + 5x + 2 & = 0 \\ \\ b^2 - 4ac & = (5)^2 - 4(p)(2) \\ & = 25 - 8p \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{No real roots}] \\ 25 - 8p & < 0 \\ -8p & < -25 \\ p & > {-25 \over -8} \\ p & > {25 \over 8} \end{align}

\begin{align} x^2 + 3x + q & = 5x - 3 \\ x^2 - 2x + (q + 3) & = 0 \\ \\ b^2 - 4ac & = (-2)^2 - 4(1)(q + 3) \\ & = 4 - 4(q + 3) \\ & = 4 - 4q - 12 \\ & = - 4q - 8 \\ \\ b^2 - 4ac & = 0 \phantom{000000} [\text{Equal roots}] \\ -4q - 8 & = 0 \\ -4q & = 8 \\ q & = {8 \over -4} \\ q & = -2 \end{align}

I think the question should be phrased as 'has real roots' instead of 'has no real roots'

\begin{align} 5cx^2 & \phantom{.} + 20x + 1 - c = 0 \\ \\ b^2 - 4ac & = (20)^2 - 4(5c)(1 - c) \\ & = 400 - 20c(1 - c) \\ & = 400 - 20c + 20c^2 \\ & = 20c^2 - 20c + 400 \\ & = 20(c^2 - c + 20) \\ & = 20 \left[ \left(c - {1 \over 2}\right)^2 - \left(1 \over 2\right)^2 + 20 \right] \phantom{000000} [\text{Complete the square}] \\ & = 20 \left[ \left(c - {1 \over 2}\right)^2 + 19{3 \over 4} \right] \\ & = 20 \left(c - {1 \over 2}\right)^2 + 395 \\ \\ \text{For all real values of } & c, \phantom{.} \left(c - {1 \over 2}\right)^2 \ge 0 \text{ and } 20 \left(c - {1 \over 2}\right)^2 + 395 \ge 395 \\ \\ \therefore \text{Equation has real ro} & \text{ots for all real values of } c \end{align}

\begin{align} x^2 + ax & = 4 - 2a \\ x^2 + ax + 2a - 4 & = 0 \\ \\ b^2 - 4ac & = (a)^2 - 4(1)(2a - 4) \\ & = a^2 - 4(2a - 4) \\ & = a^2 - 8a + 16 \\ & = a^2 - 2(a)(4) + 4^2 \\ & = (a - 4)^2 \\ \\ \text{For all real values of } & a, \phantom{.} (a - 4)^2 \ge 0 \\ \\ \therefore \text{Equation has real ro} & \text{ots for all real values of } a \end{align}

\begin{align} kx^2 - & \phantom{.} 7x + 5 \\ \\ \\ \text{Condition #1: } & k > 0 \\ \\ \\ b^2 - 4ac & = (-7)^2 - 4(k)(5) \\ & = 49 - 20k \\ \\ b^2 - 4ac & < 0 \\ 49 - 20k & < 0 \\ -20k & < -49 \\ k & > {-49 \over -20} \\ \text{Condition #2: } k & > {49 \over 20} \\ \\ \\ \therefore k & > {49 \over 20} \end{align}

\begin{align} 2x^2 + & \phantom{.} 5x + k \\ \\ \\ \text{Condition #1: } & 2 > 0 \\ \\ \\ b^2 - 4ac & = (5)^2 - 4(2)(k) \\ & = 25 - 8k \\ \\ b^2 - 4ac & < 0 \\ 25 - 8k & < 0 \\ -8k & < - 25 \\ k & > {-25 \over -8} \\ \text{Condition #2: } k & > {25 \over 8} \\ \\ \\ \therefore k & > 3{1 \over 8} \\ \\ \text{Smallest } & \text{integer} = 4 \end{align}

\begin{align} x + 2y & = 5 \\ x & = 5 - 2y \phantom{00} \text{--- (1)} \\ \\ x^2 - 3y + x & = 9 \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ (5 - 2y)^2 - 3y + 5 - 2y & = 9 \\ (5)^2 - 2(5)(2y) + (2y)^2 - 5y + 5 & = 9 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ 25 - 20y + 4y^2 - 5y + 5 & = 9 \\ 4y^2 - 25y + 30 & = 9 \\ 4y^2 - 25y + 21 & = 0 \\ (y - 1)(4y - 21) & = 0 \end{align} \begin{align} y - 1 & = 0 && \text{ or } & 4y - 21 & = 0 \\ y & = 1 &&& 4y & = 21 \\ & &&& y & = {21 \over 4} \\ \\ \text{Substitute } & \text{into (1),} &&& \text{Substitute } & \text{into (1),} \\ x & = 5 - 2(1) &&& x & = 5 - 2\left(21 \over 4\right) \\ x & = 3 &&& x & = -{11 \over 2} \end{align}
$$ x = 3, y = 1 \text{ or } x = -{11 \over 2}, y = {21 \over 4} $$

\begin{align} x^2 - 2x + 4y & = 5 \phantom{00} \text{--- (1)} \\ \\ y & = {1 \over 2}x + {5 \over 4} \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ x^2 - 2x + 4 \left({1 \over 2}x + {5 \over 4}\right) & = 5 \\ x^2 - 2x + 2x + 5 & = 5 \\ x^2 & = 0 \\ x & = 0 \\ \\ \text{Substitute } & x = 0 \text{ into (2),} \\ y & = {1 \over 2}(0) + {5 \over 4} \\ y & = {5 \over 4} \\ \\ \\ \therefore x & = 0, y = {5 \over 4} \end{align}

\begin{align} y & = 3x + 6 \phantom{00} \text{--- (1)} \\ \\ y & = -3x^2 + 7x + 1 \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 3x + 6 & = -3x^2 + 7x + 1 \\ 0 & = -3x^2 + 4x - 5 \\ \\ b^2 - 4ac & = (4)^2 - 4(-3)(-5) \\ & = -44 < 0 \\ \\ \implies \text{Equation } & 0 = -3x^2 + 4x -5 \text{ has no real roots} \\ \\ \therefore \text{Line and } & \text{curve do not intersect each other} \end{align}

(i)

\begin{align} x + y & = 7 \\ y & = 7 - x \phantom{00} \text{--- (1)} \\ \\ x^2 + xy - 2y^2 & = 10 \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ x^2 + x(7 - x) - 2(7 - x)^2 & = 10 \\ x^2 + 7x - x^2 - 2[(7)^2 - 2(7)(x) + (x)^2] & = 10 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ 7x - 2(49 - 14x + x^2) & = 10 \\ 7x - 98 + 28x - 2x^2 & = 10 \\ -2x^2 + 35x - 98 & = 10 \\ -2x^2 + 35x - 108 & = 0 \\ 2x^2 - 35x + 108 & = 0 \\ (x - 4)(2x - 27) & = 0 \end{align} \begin{align} x - 4 & = 0 && \text{ or } & 2x - 27 & = 0 \\ x & = 4 &&& 2x & = 27 \\ & &&& x & = {27 \over 2} \\ \\ \text{Substitute } & \text{into (1),} &&& \text{Substitute } & \text{into (1),} \\ y & = 7 - 4 &&& y & = 7 - {27 \over 2} \\ y & = 3 &&& y & = -{13 \over 2} \end{align}
\begin{align} x = 4, y = 3 \text{ or } x = {27 \over 2}, y = -{13 \over 2} \end{align}

(ii)

\begin{align} (4, 3), \left(13{1 \over 2}, -6{1 \over 2}\right) \end{align}

\begin{align} {1 \over x} + {1 \over y} & = {1 \over 4} \\ {y \over xy} + {x \over xy} & = {1 \over 4} \\ {y + x \over xy} & = {1 \over 4} \\ 4(y + x) & = xy \\ 4y + 4x & = xy \phantom{00} \text{--- (1)} \\ \\ x - y & = 4 \\ x & = y + 4 \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ 4y + 4(y + 4) & = (y + 4)y \\ 4y + 4y + 16 & = y^2 + 4y \\ 0 & = y^2 - 4y - 16 \\ \\ y & = { -b \pm \sqrt{b^2 - 4ac} \over 2a} \\ & = {- (-4) \pm \sqrt{(-4)^2 - 4(1)(-16)} \over 2(1)} \\ & = {4 \pm \sqrt{80} \over 2} \\ & = 6.472 \text{ or } -2.472 \end{align}
\begin{align} \text{Substitute } & y = 6.472 \text{ into (2),} &&& \text{Substitute } & y = -2.472 \text{ into (2),} \\ x & = 6.472 + 4 &&& x & = -2.472 + 4 \\ x & = 10.472 &&& x & = 1.528 \end{align}
\begin{align} \text{Coordinates: } (10.5, 6.47), (1.53, -2.47) \end{align}

(i)

\begin{align} y & = x + 2h \phantom{00} \text{--- (1)} \\ \\ y & = x^2 - 5x \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ x + 2h & = x^2 - 5x \\ 0 & = x^2 - 6x - 2h \\ \\ b^2 - 4ac & = (-6)^2 - 4(1)(-2h) \\ & = 36 - 4(-2h) \\ & = 36 + 8h \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{No intersection} \rightarrow \text{No real roots}] \\ 36 + 8h & < 0 \\ 8h & < -36 \\ h & < {-36 \over 8} \\ h & < -{9 \over 2} \end{align}

(ii) If the line is tangent to the curve, then the line meets the curve only once

$$ h = -{9 \over 2}$$

\begin{align} y & = 2px + 9p \phantom{00} \text{--- (1)} \\ \\ y & = x^2 - 4px + 9(p^2 + 1) \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 2px + 9p & = x^2 - 4px + 9(p^2 + 1) \\ 2px + 9p & = x^2 - 4px + 9p^2 + 9 \\ 0 & = x^2 - 6px + 9p^2 - 9p + 9 \\ \\ b^2 - 4ac & = (-6p)^2 - 4(1)(9p^2 - 9p + 9) \\ & = 36p^2 - 4(9p^2 - 9p + 9) \\ & = 36p^2 - 36p^2 + 36p - 36 \\ & = 36p - 36 \\ \\ b^2 - 4ac & \ge 0 \phantom{000000} [\text{Intersects once or twice}] \\ 36p - 36 & \ge 0 \\ 36p & \ge 36 \\ p & \ge {36 \over 36} \\ p & \ge 1 \end{align}

\begin{align} y & = 4x^2 + qx + 12 - q \\ \\ \text{Let } & y = 0, \\ 0 & = 4x^2 + qx + 12 - q \\ \\ b^2 - 4ac & = (q)^2 - 4(4)(12 - q) \\ & = q^2 - 16(12 - q) \\ & = q^2 - 192 + 16q \\ & = q^2 + 16q - 192 \\ \\ b^2 - 4ac & = 0 \phantom{000000} [\text{Touches - meets once}] \\ q^2 + 16q - 192 & = 0 \\ (q + 24)(q - 8) & = 0 \\ \\ q & = -24 \text{ or } 8 \end{align}

\begin{align} \text{Let } x \text{ and } y \text{ denote } & \text{the radius of each respective circle} \\ \\ \text{Circumference} & = 2\pi r \\ \\ 2\pi x + 2\pi y & = 98 \pi \\ x + y & = 49 \phantom{0000000000} [\text{Divide by } 2\pi] \\ y & = 49 - x \phantom{00} \text{--- (1)} \\ \\ \\ \text{Area of circle} & = \pi r^2 \\ \\ \pi (x)^2 + \pi (y)^2 & = 1421 \pi \\ \pi x^2 + \pi y^2 & = 1421 \pi \\ x^2 + y^2 & = 1421 \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ x^2 + (49 - x)^2 & = 1421 \\ x^2 + (49)^2 - 2(49)(x) + (x)^2 & = 1421 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ x^2 + 2401 - 98x + x^2 & = 1421 \\ 2x^2 - 98x + 980 & = 0 \\ x^2 - 49x + 490 & = 0 \\ (x - 14)(x - 35) & = 0 \end{align} \begin{align} x - 14 & = 0 && \text{ or } & x - 35 & = 0 \\ x & = 14 &&& x & = 35 \\ \\ \text{Subsitute} & \text{ into (1),} &&& \text{Subsitute} & \text{ into (1),} \\ y & = 49 - 14 &&& y & = 49 - 35 \\ y & = 35 &&& y & = 14 \end{align}
\begin{align} \therefore \text{Radius of one circle is 14 cm and the other is 35 cm} \end{align}

(i)

\begin{align} R_A & = R_B + 6 \phantom{00} \text{--- (1)} \\ \\ \\ {1 \over R_T} & = {1 \over R_A} + {1 \over R_B} \\ {1 \over 1.6} & = {R_B \over R_A R_B} + {R_A \over R_A R_B} \\ {5 \over 8} & = {R_B + R_A \over R_A R_B} \\ 5 R_A R_B & = 8(R_B + R_A) \\ 5 R_A R_B & = 8R_B + 8R_A \phantom{00} \text{--- (2)} \end{align}

(ii)

\begin{align} \text{Substitute } & \text{(1) into (2),} \\ 5 (R_B + 6)R_B & = 8R_B + 8(R_B + 6) \\ 5 R_B (R_B + 6) & = 8R_B + 8R_B + 48 \\ 5 R_B^2 + 30 R_B & = 16R_B + 48 \\ 5 R_B^2 + 14R_B - 48 & = 0 \\ (R_B - 2)(5 R_B + 24) & = 0 \end{align} \begin{align} R_B - 2 & = 0 && \text{ or } & 5R_B + 24 & = 0 \\ R_B & = 2 &&& 5R_B & = -24 \\ & &&& R_B & = -{24 \over 5} \text{ (Reject, since } R_B \ge 0) \\ \\ \text{Subsitute} & \text{ into (1),} \\ R_A & = 2 + 6 \\ & = 8 \end{align}
\begin{align} \therefore R_A = 8 \text{ ohms}, R_B = 2 \text{ ohms} \end{align}

(i)

\begin{align} x^2 - px + p + 3 & = 0 \\ \\ b^2 - 4ac & = (-p)^2 - 4(1)(p + 3) \\ & = p^2 - 4(p + 3) \\ & = p^2 - 4p - 12 \\ \\ b^2 - 4ac & > 0 \\ p^2 - 4p - 12 & > 0 \\ (p + 2)(p - 6) & > 0 \end{align}

$$ p < -2 \text{ or } p > 6 $$

(ii)

$$ p = - 2 \text{ or } 6 \phantom{000000} [\text{so that } b^2 - 4ac = 0] $$

(i)

\begin{align} y & = kx - 2 \phantom{00} \text{--- (1)} \\ \\ y & = x^2 + 3x + 7 \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ kx - 2 & = x^2 + 3x + 7 \\ 0 & = x^2 + 3x - kx + 9 \\ 0 & = x^2 + (3 - k)x + 9 \\ \\ b^2 - 4ac & = (3 - k)^2 - 4(1)(9) \\ & = (3)^2 - 2(3)(k) + (k)^2 - 36 \phantom{000000} [(a - b)^2 = a^2 - 2ab + b^2] \\ & = 9 - 6k + k^2 - 36 \\ & = k^2 - 6k - 27 \\ \\ b^2 - 4ac & < 0 \phantom{000000} [\text{No intersection, thus no real roots}] \\ k^2 - 6k - 27 & < 0 \\ (k + 3)(k - 9) & < 0 \end{align}

$$ -3 < k < 9 $$

(ii) If the line is tangent to the curve, the line meets the curve only once

$$ p = - 3 \text{ or } 9 \phantom{000000} [\text{so that } b^2 - 4ac = 0] $$

\begin{align} y & = kx + 6 \phantom{00} \text{--- (1)} \\ \\ 2x^2 & = xy + 3 \phantom{00} \text{--- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 2x^2 & = x(kx + 6) + 3 \\ 2x^2 & = kx^2 + 6x + 3 \\ 0 & = kx^2 - 2x^2 + 6x + 3 \\ 0 & = (k - 2)x^2 + 6x + 3 \\ \\ b^2 - 4ac & = (6)^2 - 4(k - 2)(3) \\ & = 36 - 12(k - 2) \\ & = 36 - 12k + 24 \\ & = 60 - 12k \\ \\ b^2 - 4ac & \ge 0 \phantom{000000} [\text{Line can meet curve once or twice}] \\ 60 - 12k & \ge 0 \\ -12k & \ge -60 \\ k & \le {-60 \over -12} \\ k & \le 5 \end{align}