(a)

\begin{align} 3x^2 + 9x & = 1 \\ 3x^2 + 9x - 1 & = 0 \\ \\ [a = 3, b & = 9, c = -1] \\ \\ \text{Sum of roots} & = -{b \over a} \\ & = -{9 \over 3} \\ & = -3 \\ \\ \text{Product of roots} & = {c \over a} \\ & = {-1 \over 3} \\ & = -{1 \over 3} \end{align}

(b)

\begin{align} 4x + 2x^2 & = 3x^2 + 2 \\ 0 & = 3x^2 - 2x^2 - 4x + 2 \\ 0 & = x^2 - 4x + 2 \\ \\ [a = 1, b & = -4, c = 2] \\ \\ \text{Sum of roots} & = -{b \over a} \\ & = -{-4 \over 1} \\ & = 4 \\ \\ \text{Product of roots} & = {c \over a} \\ & = {2 \over 1} \\ & = 2 \end{align}

\begin{align} x^2 + 3x + 1 & = 0 \\ \\ [a = 1, b & = 3, c = 1] \\ \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{3 \over 1} \\ \alpha + \beta & = -3 \\ \\ \text{Product of roots} & = {c \over a} \\ \alpha \beta & = {1 \over 1} \\ \alpha \beta & = 1 \end{align}

(i)

\begin{align} {2 \over \alpha} + {2 \over \beta} & = {2\beta \over \alpha \beta} + {2\alpha \over \alpha \beta} \\ & = {2\beta + 2\alpha \over \alpha \beta} \\ & = {2(\beta + \alpha) \over \alpha \beta} \\ & = {2(-3) \over 1} \phantom{00000} [\text{Substitute in values found earlier}] \\ & = -6 \end{align}

(ii)

\begin{align} (2\alpha - 1)(2\beta - 1) & = 4\alpha \beta - 2\alpha - 2\beta + 1 \\ & = 4(1) - 2(\alpha + \beta) + 1 \\ & = 4 - 2(-3) + 1 \\ & = 11 \end{align}

\begin{align} 40x^2 - 138x + 119 & = 0 \\ \\ [a = 40, b & = -138, c = 119] \\ \\ \\ \text{Sum of roots} & = -{b \over a} \\ & = -{-138 \over 40} \\ & = 3.45 \\ \\ \text{Product of roots} & = {c \over a} \\ & = {119 \over 40} \\ & = 2.975 \\ \\ \\ \text{Since the solutions (or roots) of the} & \text{ equation are the heights of the men,} \\ \\ \text{Average height} & = {\text{Sum of roots} \over 2} \\ & = {3.45 \over 2} \\ & = 1.725 \text{ m} \end{align}

\begin{align} 2x^2 - x - 4 & = 0 \\ \\ [a = 2, b & = -1, c = -4] \\ \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-1 \over 2} \\ \alpha + \beta & = {1 \over 2} \\ \\ \text{Product of roots} & = {c \over a} \\ \alpha \beta & = {-4 \over 2} \\ \alpha \beta & = -2 \\ \end{align}

(i)

\begin{align} (\alpha + \beta)^2 & = (\alpha)^2 + 2(\alpha)(\beta) + (\beta)^2 \\ (\alpha + \beta)^2 & = \alpha^2 + 2\alpha \beta + \beta^2 \\ (\alpha + \beta)^2 & = \alpha^2 + \beta^2 + 2\alpha \beta \\ \left(1 \over 2\right)^2 & = \alpha^2 + \beta^2 + 2(-2) \phantom{00000} [\text{Substitute in values found earlier}] \\ {1 \over 4} & = \alpha^2 + \beta^2 - 4 \\ {17 \over 4} & = \alpha^2 + \beta^2 \end{align}

(ii)

\begin{align} (\alpha - \beta)^2 & = (\alpha)^2 - 2\alpha \beta + (\beta)^2 \\ & = \alpha^2 - 2\alpha \beta + \beta^2 \\ & = (\alpha^2 + \beta^2) - 2\alpha \beta \\ & = {17 \over 4} - 2(-2) \phantom{00000} [\text{Substitute in values found earlier}] \\ & = {33 \over 4} \\ \\ \alpha - \beta & = \pm \sqrt{33 \over 4} \\ & = \pm {\sqrt{33} \over \sqrt{4}} \\ & = \pm {\sqrt{33} \over 2} \end{align}

(i)

\begin{align} \text{Let } \alpha \text{ denote the } & \text{larger root.} \\ \text{Let } \beta \text{ denote the } & \text{smaller root.} \\ \\ x^2 - 4x + c & = 0 \\ \\ [a = 1, b & = -4] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-4 \over 1} \\ \alpha + \beta & = 4 \phantom{000} \text{ --- (1)} \\ \\ \alpha - \beta & = 2 \\ \alpha & = \beta + 2 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(2) into (1),} \\ \\ (\beta + 2) + \beta & = 4 \\ 2\beta + 2 & = 4 \\ 2\beta & = 2 \\ \beta & = {2 \over 2} \\ & = 1 \\ \\ \text{Substitute } \beta & = 1 \text{ into (2),} \\ \\ \alpha & = 1 + 2 \\ & = 3 \end{align}

(ii)

\begin{align} \text{Product of roots} & = {c \over a} \\ \alpha \beta & = {c \over 1} \\ (3)(1) & = c \\ 3 & = c \end{align}

(a)

\begin{align} \text{Sum of roots} & = 2 + 5 \\ & = 7 \\ \\ \text{Product of roots} & = (2)(5) \\ & = 10 \\ \\ x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} & = 0 \\ x^2 - (7)x + (10) & = 0 \\ \\ x^2 - 7x + 10 & = 0 \end{align}

(b)

\begin{align} \text{Sum of roots} & = -1 + 3 \\ & = 2 \\ \\ \text{Product of roots} & = (-1)(3) \\ & = -3 \\ \\ x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} & = 0 \\ x^2 - (2)x + (-3) & = 0 \\ x^2 - 2x - 3 & = 0 \end{align}

\begin{align} 2x^2 - 4x + 5 & = 0 \\ \\ [a = 2, b & = -4, c = 5] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-4 \over 2} \\ \alpha + \beta & = 2 \\ \\ \text{Product of roots} & = {c \over a} \\ \alpha \beta & = {5 \over 2} \end{align}

(i)

\begin{align} \text{Sum of roots} & = (\alpha - 1) + (\beta - 1) \\ & = \alpha + \beta - 2 \\ & = (\alpha + \beta) - 2 \\ & = (2) - 2 \\ & = 0 \\ \\ \text{Product of roots} & = (\alpha - 1)(\beta - 1) \\ & = \alpha \beta - \alpha - \beta + 1 \\ & = \alpha \beta - (\alpha + \beta) + 1 \\ & = {5 \over 2} - (2) + 1 \\ & = {3 \over 2} \\ \\ \\ x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} & = 0 \\ x^2 - (0)x + {3 \over 2} & = 0 \\ x^2 + {3 \over 2} & = 0 \\ 2x^2 + 3 & = 0 \end{align}

(ii)

\begin{align} \text{Sum of roots} & = 2\alpha + 2\beta \\ & = 2(\alpha + \beta) \\ & = 2(2) \\ & = 4 \\ \\ \text{Product of roots} & = (2\alpha)(2\beta) \\ & = 4\alpha \beta \\ & = 4\left(5 \over 2\right) \\ & = 10 \\ \\ \\ x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} & = 0 \\ x^2 - (4)x + 10 & = 0 \\ x^2 - 4x + 10 & = 0 \end{align}

(a)

\begin{align} x^2 + px + q & = 0 \\ \\ [a = 1, b & = p, c = q] \\ \\ \\ \text{Sum of roots} & = -{b \over a} \\ & = -{p \over 1} \\ & = -p \\ \\ \text{Product of roots} & = {c \over a} \\ & = {q \over 1} \\ & = q \end{align}

(b)

\begin{align} \text{Sum of roots} & = -{p \over 1} \\ \alpha + 4\alpha & = -p \\ 5\alpha & = -p \\ \alpha & = -{p \over 5} \phantom{000} \text{ --- (1)} \\ \\ \text{Product of roots} & = {q \over 1} \\ (\alpha)(4\alpha) & = q \\ 4\alpha^2 & = q \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 4 \left(-{p \over 5}\right)^2 & = q \\ \left(4 \over 1\right) \left( p^2 \over 25 \right) & = q \\ {4p^2 \over 25} & = q \\ 4p^2 & = 25q \text{ (Shown)} \end{align}

\begin{align} 2x^2 - x - 2 & = 0 \\ \\ [a = 2, b & = -1, c = -2] \\ \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-1 \over 2} \\ \alpha + \beta & = {1 \over 2} \\ \\ \text{Product of roots} & = {c \over a} \\ \alpha \beta & = {-2 \over 2} \\ \alpha \beta & = -1 \end{align}

(i)

\begin{align} \alpha^2 + \beta^2 & = (\alpha + \beta)^2 - 2\alpha \beta \\ & = \left(1 \over 2\right)^2 - 2(-1) \\ & = {1 \over 4} + 2 \\ & = {9 \over 4} \end{align}

(ii)

\begin{align} {\beta \over \alpha} + {\alpha \over \beta} & = {\beta(\beta) \over \alpha \beta} + {\alpha (\alpha) \over \alpha \beta} \\ & = {\beta^2 + \alpha^2 \over \alpha \beta} \\ & = { {9 \over 4} \over -1} \\ & = -{9 \over 4} \end{align}

(iii)

\begin{align} \alpha^4 + \beta^4 & = (\alpha^2)^2 + (\beta^2)^2 \\ & = (\alpha^2 + \beta^2)^2 - 2 \alpha^2 \beta^2 \\ & = \left(9 \over 4\right)^2 - 2 (\alpha \beta)^2 \\ & = {81 \over 16} - 2(-1)^2 \\ & = {49 \over 16} \end{align}

(i)

\begin{align} \text{Let } \alpha \text{ denote the } & \text{larger root.} \\ \text{Let } \beta \text{ denote the } & \text{smaller root.} \\ \\ \alpha - \beta & = 3 \\ \alpha & = 3 + \beta \phantom{000} \text{ --- (1)} \\ \\ x^2 & = (k - 1)x + k \\ x^2 - (k - 1) - k & = 0 \\ x^2 + (1 - k) - k & = 0 \\ \\ [a = 1, b & = 1 - k, c = -k] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{1 - k \over 1} \\ \alpha + \beta & = -(1 - k) \\ (3 + \beta) + \beta & = -1 + k \\ 3 + 2\beta + 1 & = k \\ 4 + 2\beta & = k \phantom{000} \text{ --- (2)} \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha)(\beta) & = {-k \over 1} \\ (3 + \beta)(\beta) & = -k \phantom{000} \text{ --- (3)} \\ \\ \text{Substitute } & \text{(2) into (3),} \\ (3 + \beta)(\beta) & = - (4 + 2\beta) \\ 3\beta + \beta^2 & = -4 - 2 \beta \\ \beta^2 + 5\beta + 4 & = 0 \\ (\beta + 1)(\beta + 4) & = 0 \end{align} \begin{align} \beta + 1 & = 0 &\text{or }\phantom{00} \beta + 4 & = 0 \\ \beta & = - 1 & \beta & = - 4 \\ \\ \text{Substitute } & \text{into (1),} \\ \alpha & = 3 + (-1) & \alpha & = 3 + (-4) \\ & = 2 \text{ (Reject)} & & = -1 \end{align} $$ \therefore \alpha = -1, \beta = -4 $$

(ii)

\begin{align} \text{Substitute } \beta & = -4 \text{ into (2),} \\ 4 + 2(-4) & = k \\ 4 - 8 & = k \\ -4 & = k \end{align}

\begin{align} 3x^2 - 3kx + k - 6 & = 0 \\ \\ [a = 3, b & = -3k, c = k - 6] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-3k \over 3} \\ \alpha + \beta & = -(-k) \\ \alpha + \beta & = k \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha)(\beta) & = {k - 6 \over 3} \\ \alpha \beta & = {k - 6 \over 3} \\ \\ \alpha^2 + \beta^2 & = (\alpha + \beta)^2 - 2\alpha \beta \\ {20 \over 3} & = (k)^2 - 2\left(k - 6 \over 3\right) \\ {20 \over 3} & = k^2 - {2(k - 6) \over 3} \\ 20 & = 3k^2 - 2(k - 6) \\ 20 & = 3k^2 - 2k + 12 \\ 0 & = 3k^2 - 2k - 8 \\ 0 & = (3k + 4)(k - 2) \end{align} \begin{align} 3k + 4 & = 0 &\text{or }\phantom{00} k - 2 & = 0 \\ 3k & = -4 & k & = 2 \\ k & = -{4 \over 3} \end{align}

\begin{align} 2x^2 - 3x + 6 & = 0 \\ \\ [a = 2, b & = -3, c = 6] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-3 \over 2} \\ \alpha + \beta & = {3 \over 2} \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha)(\beta) & = {6 \over 2} \\ \alpha \beta & = 3 \end{align}

(i)

\begin{align} \text{Sum of roots} & = {2\alpha \over \beta} + {2\beta \over \alpha} \\ & = {2\alpha (\alpha) \over \alpha \beta} + {2\beta (\beta) \over \alpha \beta} \\ & = {2\alpha^2 \over \alpha \beta} + {2\beta^2 \over \alpha \beta} \\ & = {2\alpha^2 + 2\beta^2 \over \alpha \beta} \\ & = {2 (\alpha^2 + \beta^2) \over \alpha \beta} \\ & = {2 [(\alpha + \beta)^2 - 2\alpha \beta] \over \alpha \beta} \\ & = {2(\alpha + \beta)^2 - 4\alpha \beta \over \alpha \beta} \\ & = {2\left(3 \over 2\right)^2 - 4(3) \over (3) } \\ & = -2.5 \\ \\ \text{Product of roots} & = \left(2\alpha \over \beta\right)\left(2\beta \over \alpha\right) \\ & = {4\alpha \beta \over \alpha \beta} \\ & = 4 \\ \\ \\ x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} & = 0 \\ x^2 - (-2.5)x + 4 & = 0 \\ x^2 + 2.5x + 4 & = 0 \\ 2x^2 + 5x + 8 & = 0 \end{align}

(ii)

\begin{align} \text{Sum of roots} & = (3\alpha + \beta) + (\alpha + 3\beta) \\ & = 4\alpha + 4\beta \\ & = 4(\alpha + \beta) \\ & = 4\left(3 \over 2\right) \\ & = 6 \\ \\ \text{Product of roots} & = (3\alpha + \beta)(\alpha + 3\beta) \\ & = 3\alpha^2 + 9\alpha \beta + \alpha \beta + 3\beta^2 \\ & = 3\alpha^2 + 3\beta^2 + 10 \alpha\beta \\ & = 3(\alpha^2 + \beta^2) + 10 \alpha \beta \\ & = 3[ (\alpha + \beta)^2 - 2\alpha \beta] + 10 \alpha \beta \\ & = 3(\alpha + \beta)^2 - 6 \alpha \beta + 10 \alpha \beta \\ & = 3(\alpha + \beta)^2 + 4\alpha \beta \\ & = 3\left(3 \over 2\right)^2 + 4(3) \\ & = {75 \over 4} \\ \\ x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} & = 0 \\ x^2 - (6)x + {75 \over 4} & = 0 \\ 4x^2 - 24x + 75 & = 0 \end{align}

\begin{align} 2x^2 & = 8x + 3 \\ 2x^2 - 8x - 3 & = 0 \\ \\ [a = 2, b & = -8, c = -3] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-8 \over 2} \\ \alpha + \beta & = 4 \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha)(\beta) & = {-3 \over 2} \\ \alpha \beta & = -{3 \over 2} \end{align}

(i)

\begin{align} \text{Sum of roots} & = {1 \over \alpha^2} + {1 \over \beta^2} \\ & = {\beta^2 \over \alpha^2 \beta^2} + {\alpha^2 \over \alpha^2 \beta^2} \\ & = {\beta^2 + \alpha^2 \over \alpha^2 \beta^2} \\ & = {(\alpha + \beta)^2 - 2\alpha\beta \over (\alpha \beta)^2} \\ & = {(4)^2 - 2 \left(-{3 \over 2}\right) \over \left(-{3 \over 2}\right)^2} \\ & = {76 \over 9} \\ \\ \text{Product of roots} & = \left(1 \over \alpha^2\right)\left(1 \over \beta^2\right) \\ & = {1 \over \alpha^2 \beta^2} \\ & = {1 \over (\alpha \beta)^2} \\ & = {1 \over \left(-{3 \over 2}\right)^2 } \\ & = {4 \over 9} \\ \\ x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} & = 0 \\ x^2 - {76 \over 9}x + {4 \over 9} & = 0 \\ 9x^2 - 76x + 4 & = 0 \end{align}

(ii)

\begin{align} \text{Sum of roots} & = \alpha^2 \beta + \alpha \beta^2 \\ & = \alpha \beta (\alpha + \beta) \\ & = -{3 \over 2} (4) \\ & = - 6 \\ \\ \text{Product of roots} & = (\alpha^2 \beta)(\alpha \beta^2) \\ & = \alpha^3 \beta^3 \\ & = (\alpha \beta)^3 \\ & = \left(-{3 \over 2}\right)^3 \\ & = -{27 \over 8} \\ \\ \\ x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} & = 0 \\ x^2 - (-6)x + \left(-{27 \over 8}\right) & = 0 \\ x^2 + 6x - {27 \over 8} & = 0 \\ 8x^2 + 48x - 27 & = 0 \end{align}

(iii)

\begin{align} \text{Sum of roots} & = \alpha - \beta + \beta - \alpha \\ & = 0 \\ \\ \text{Product of roots} & = (\alpha - \beta)(\beta - \alpha) \\ & = \alpha \beta - \alpha^2 - \beta^2 + \alpha \beta \\ & = 2\alpha \beta - (\alpha^2 + \beta^2) \\ & = 2\left(-{3 \over 2}\right) - [(\alpha + \beta)^2 - 2\alpha \beta] \\ & = -3 - \left[ (4)^2 - 2\left(-{3 \over 2}\right) \right] \\ & = - 22 \\ \\ x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} & = 0 \\ x^2 - (0)x + (-22) & = 0 \\ x^2 - 22 & = 0 \end{align}

(i)

\begin{align} \text{Let } \alpha \text{ denote the } & \text{breadth of the rectangle}. \\ \text{Let } \beta \text{ denote the } & \text{length of the rectangle}. \\ \\ 2x^2 - 71x + 615 & = 0 \\ \\ [a = 2, b & = -71, c = 615] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-71 \over 2} \\ \alpha + \beta & = 35.5 \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha)(\beta) & = {615 \over 2} \\ \alpha \beta & = 307.5 \\ \\ \text{Area} & = \text{Breadth} \times \text{Length} \\ & = \alpha \times \beta \\ & = \alpha \beta \\ & = 307.5 \text{ cm}^2 \\ \\ \text{Perimeter} & = 2 \times \text{Breadth} + 2 \times \text{Length} \\ & = 2 \times \alpha + 2 \times \beta \\ & = 2\alpha + 2\beta \\ & = 2(\alpha + \beta) \\ & = 2(35.5) \\ & = 71 \text{ cm} \end{align}

(ii)

\begin{align} & \text{Since Area of parallelogram = Base } \times \text{ Height, the result for area is valid} \\ & \text{Since the height of parallelogram is not equals to it's length, the result for perimeter is not valid} \end{align}

\begin{align} \text{Let } \alpha \text{ denote the } & \text{smaller root}. \\ \text{The other root is } & 2 \alpha. \\ \\ kx^2 + (k - 1)x + 2k + 3 & = 0 \\ \\ [a = k, b & = k - 1, c = 2k + 3] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + 2\alpha & = -{(k - 1) \over k} \\ 3\alpha & = -{k - 1 \over k} \\ \alpha & = -{k - 1 \over 3k} \phantom{000} \text{ --- (1)} \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha)(2\alpha) & = {2k + 3 \over k} \\ 2\alpha^2 & = {2k + 3 \over k} \phantom{000} \text{ --- (2)} \\ \\ \\ \text{Substitute } & \text{(1) into (2),} \\ 2 \left(-{k - 1 \over 3k}\right)^2 & = {2k + 3 \over k} \\ 2 \left[ (k - 1)^2 \over (3k)^2 \right] & = {2k + 3 \over k} \\ {2(k - 1)^2 \over (3k)^2} & = {2k + 3 \over k} \\ 2k(k - 1)^2 & = (3k)^2 (2k + 3) \\ 2k (k^2 - 2k + 1) & = 9k^2 (2k + 3) \\ 2k^3 - 4k^2 + 2k & = 18k^3 + 27k^2 \\ 0 & = 16k^3 + 31k^2 - 2k \\ 0 & = k(16k^2 + 31k - 2) \\ 0 & = k(16k - 1)(k + 2) \end{align}
\begin{align} k & = 0 \text{ (Reject)} & \text{or }\phantom{00} 16k - 1 & = 0 & \text{or }\phantom{00} k + 2 & = 0 \\ & & 16k & = 1 & k & = - 2 \\ & & k & = {1 \over 16} \text{ (Reject)} \end{align}

\begin{align} x^2 - 3x - 2 & = 0 \\ \\ [a = 1, b & = -3, c = - 2] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-3 \over 1} \\ \alpha + \beta & = 3 \\ \\ \text{Product of roots} & = {c \over a} \\ \alpha \beta & = {-2 \over 1} \\ \alpha \beta & = -2 \\ \\ \\ x^2 - 6x + p & = 0 \\ \\ [a = 1, b & = - 6, c = p] \\ \\ \text{Sum of roots} & = -{b \over a} \\ {k \over \alpha} + {k \over \beta} & = -{-6 \over 1} \\ {k\beta \over \alpha \beta} + {k\alpha \over \alpha \beta} & = 6 \\ {k\beta + k\alpha \over \alpha \beta} & = 6 \\ {k(\beta + \alpha) \over \alpha \beta} & = 6 \\ {k(3) \over -2} & = 6 \\ 3k & = -2(6) \\ & = -12 \\ k & = {-12 \over 3} \\ & = -4 \\ \\ \text{Product of roots} & = {c \over a} \\ \left(k \over \alpha\right)\left(k \over \beta\right) & = {p \over 1} \\ {k^2 \over \alpha \beta} & = p \\ {(-4)^2 \over (-2)} & = p \\ {16 \over -2} & = p \\ -8 & = p \end{align}

(a)

\begin{align} \text{Let } \alpha \text{ denote the } & \text{smaller root.} \\ \text{The other root is } & 2\alpha. \\ \\ 3x^2 + kx + 96 & = 0 \\ \\ [a = 3, b & = k, c = 96] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + 2\alpha & = -{k \over 3} \\ 3 \alpha & = -{k \over 3} \\ \alpha & = -{k \over 9} \phantom{000} \text{ --- (1)} \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha)(2\alpha) & = {96 \over 3} \\ 2\alpha^2 & = 32 \\ \alpha^2 & = {32 \over 2} \\ \alpha^2 & = 16 \phantom{000} \text{ --- (2)} \\ \\ \text{Substitute } & \text{(1) into (2),} \\ \\ \left(-{k \over 9}\right)^2 & = 16 \\ {k^2 \over 81} & = 16 \\ k^2 & = 81(16) \\ & = 1296 \\ k & = \pm \sqrt{1296} \\ & = \pm 36 \\ \\ \\ \text{Since both roots are positive, } & \text{sum of roots must be positive}. \\ \\ \text{For sum of roots} = -{k \over 9} & \text{ to be positive } \implies k = - 36 \end{align}

(b)

\begin{align} \text{Sum of roots} & = p^2 + q^2 \\ & = 13 \\ \\ \text{Product of roots} & = (p^2)(q^2) \\ & = p^2 q^2 \\ & = (pq)^2 \\ & = 6^2 \\ & = 36 \\ \\ x^2 - \text{(Sum of roots)}x + \text{(Product of roots)} & = 0 \\ x^2 - (13)x + 36 & = 0 \\ x^2 - 13x + 36 & = 0 \\ \\ \therefore \text{Quadratic equation: } & \phantom{0} x^2 - 13x + 36 = 0 \\ \\ \\ x^2 - 13x + 36 & = 0 \\ (x - 4)(x - 9) & = 0 \end{align} \begin{align} x - 4 & = 0 &\text{or }\phantom{00} x - 9 & = 0 \\ x & = 4 & x & = 9 \\ \\ p^2 & = 4 &\text{or }\phantom{0000} p^2 & = 9 \\ p & = \pm \sqrt{4} & p & = \pm \sqrt{9} \\ & = \pm 2 & & = \pm 3 \end{align}

The longest side in the triangle is the hypotenuse. The two shorter sides (base & height) join to form a right angle.

\begin{align} \text{Let } \alpha \text{ denote the } & \text{base of the right-angled triangle.} \\ \text{Let } \beta \text{ denote the } & \text{height of the right-angled triangle.} \\ \\ 5x^2 - 102x + 432 & = 0 \\ \\ [a = 5, b & = -102, c = 432] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-102 \over 5} \\ \alpha + \beta & = 20.4 \\ \\ \text{Product of roots} & = {c \over a} \\ \alpha \beta & = {432 \over 5} \\ \alpha \beta & = 86.4 \\ \\ \\ \text{Area} & = {1 \over 2} \times \text{Base} \times \text{Height} \\ & = {1 \over 2} \times \alpha \beta \\ & = {1 \over 2} \times 86.4 \\ & = 43.2 \text{ cm}^2 \\ \\ \\ \text{By Pythagora's } & \text{theorem,} \\ \text{Hypotenuse}^2 & = \alpha^2 + \beta^2 \\ & = (\alpha + \beta)^2 - 2\alpha \beta \\ & = (20.4)^2 - 2(86.4) \\ & = 243.36 \\ \\ \text{Hypotenuse} & = \pm \sqrt{243.36} \\ & = \pm 15.6 \\ & = 15.6 \text{ cm} \text{ or } -15.6 \text{ cm (Reject)} \\ \\ \text{Perimeter} & = \text{Base} + \text{Height} + \text{Hypothenuse} \\ & = \alpha + \beta + 15.6 \\ & = (\alpha + \beta) + 15.6 \\ & = 20.4 + 15.6 \\ & = 36 \text{ cm} \end{align}

(a)

\begin{align} 2x^2 + px + 24 & = 0 \\ \\ [a = 2, b & = p, c = 24] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{p \over 2} \\ \alpha + \beta & = -{1 \over 2}p \phantom{000} \text{ --- (1)} \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha) (\beta) & = {24 \over 2} \\ \alpha \beta & = 12 \phantom{000} \text{ --- (2)} \\ \\ \alpha - \beta & = 4 \\ \alpha & = \beta + 4 \phantom{000} \text{ --- (3)} \\ \\ \text{Substitute } & \text{(3) into (2),} \\ (\beta + 4)\beta & = 12 \\ \beta^2 + 4\beta & = 12 \\ \beta^2 + 4\beta - 12 & = 0 \\ (\beta + 6)(\beta - 2) & = 0 \end{align} \begin{align} \beta + 6 & = 0 &\text{or }\phantom{00} \beta - 2 & = 0 \\ \beta & = -6 & \beta & = 2 \\ \\ \text{Substitute } & \text{into (3),} \\ \alpha & = (-6) + 4 & \alpha & = 2 + 4 \\ & = -2 & & = 6 \\ \\ \text{Substitute } & \text{into (1),} \\ (-2) + (-6) & = -{1 \over 2}p & (6) + (2) & = -{1 \over 2}p \\ -8 & = -{1 \over 2}p & 8 & = -{1 \over 2}p \\ {-8 \over -{1 \over 2}} & = p & {8 \over -{1 \over 2}} & = p \\ 16 & = p \text{ (Reject)} & -16 & = p \end{align}

(b)

\begin{align} \text{Let } \alpha \text{ & } \beta \text{ denote the } & \text{roots of the equation } 3x^2 - x + 2 = 0. \\ \\ [a = 3, b & = -1, c = 2] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{-1 \over 3} \\ & = {1 \over 3} \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha)(\beta) & = {2 \over 3} \\ \alpha \beta & = {2 \over 3} \\ \\ \\ \text{For the new equation, } & \text{roots are } \alpha^2 \text{ & } \beta^2. \\ \\ \text{Sum of roots} & = \alpha^2 + \beta^2 \\ & = (\alpha + \beta)^2 - 2\alpha \beta \\ & = \left(1 \over 3\right)^2 - 2\left(2 \over 3\right) \\ & = -{11 \over 9} \\ \\ \text{Product of roots} & = (\alpha^2)(\beta^2) \\ & = \alpha^2 \beta^2 \\ & = (\alpha \beta)^2 \\ & = \left(2 \over 3\right)^2 \\ & = {4 \over 9} \\ \\ x^2 - \text{(Sum of roots)}x & + \text{(Product of roots)} = 0 \\ x^2 - \left(-{11 \over 9}\right)x + {4 \over 9} & = 0 \\ x^2 + {11 \over 9}x + {4 \over 9} & = 0 \\ 9x^2 + 11x + 4 & = 0 \end{align}

\begin{align} 4x^2 - x & + 36 = 0 \\ \\ [a = 4, b & = -1, c = 36] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha^2 + \beta^2 & = -{-1 \over 4} \\ & = {1 \over 4} \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha^2)(\beta^2) & = {36 \over 4} \\ \alpha^2 \beta^2 & = 9 \\ \alpha \beta & = \pm \sqrt{9} \\ & = \pm 3 \end{align}

(i)

\begin{align} \text{Sum of roots} & = {1 \over \alpha^2} + {1 \over \beta^2} \\ & = {\beta^2 \over \alpha^2 \beta^2} + {\alpha^2 \over \alpha^2 \beta^2} \\ & = {\beta^2 + \alpha^2 \over \alpha^2 \beta^2} \\ & = { {1 \over 4} \over 9} \\ & = {1 \over 36} \\ \\ \text{Product of roots} & = \left(1 \over \alpha^2\right) \left(1 \over \beta^2\right) \\ & = {1 \over \alpha^2 \beta^2} \\ & = {1 \over 9} \\ \\ \\ x^2 - \text{(Sum of roots)}x & + \text{(Product of roots)} = 0 \\ x^2 - {1 \over 36}x + {1 \over 9} & = 0 \\ 36x^2 - x + 4 & = 0 \end{align}

(ii)

\begin{align} \alpha^2 + \beta^2 & = (\alpha + \beta)^2 - 2\alpha \beta \\ {1 \over 4} & = (\alpha + \beta)^2 - 2(\pm 3) \end{align} \begin{align} {1 \over 4} & = (\alpha + \beta)^2 - 2(3) &\text{or }\phantom{00000} {1 \over 4} & = (\alpha + \beta)^2 - 2(-3) \\ {1 \over 4} & = (\alpha + \beta)^2 - 6 & {1 \over 4} & = (\alpha + \beta)^2 + 6 \\ {1 \over 4} + 6 & = (\alpha + \beta)^2 & {1 \over 4} - 6 & = (\alpha + \beta)^2 \\ {25 \over 4} & = (\alpha + \beta)^2 & -{23 \over 4} & = (\alpha + \beta)^2 \\ \pm \sqrt{25 \over 4} & = \alpha + \beta & \pm \sqrt{-{23 \over 4}} & = (\alpha + \beta)^2 \phantom{00} \text{ (Reject)} \\ \pm {5 \over 2} & = \alpha + \beta \end{align} $$ \text{Reject } \alpha \beta = -3 $$ \begin{align} \text{Sum of roots} & = \alpha + \beta \\ & = \pm {5 \over 2} \\ \\ \text{Product of roots} & = (\alpha)(\beta) \\ & = \alpha \beta \\ & = 3 \\ \\ \\ x^2 - \text{(Sum of roots)}x & + \text{(Product of roots)} = 0 \\ \\ \text{When SOR} & = {5 \over 2} \text{ and POR} = 3, \\ x^2 - {5 \over 2}x + 3 & = 0 \\ 2x^2 - 5x + 6 & = 0 \phantom{00} [\text{1st equation}] \\ \\ \\ \text{When SOR} & = -{5 \over 2} \text{ and POR} = 3, \\ x^2 - \left(-{5 \over 2}\right)x + 3 & = 0 \\ x^2 + {5 \over 2}x + 3 & = 0 \\ 2x^2 + 5x + 6 & = 0 \phantom{00} [\text{2nd equation}] \end{align}

(i)

\begin{align} \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{b \over a} \\ \\ \text{Product of roots} & = {c \over a} \\ \alpha \beta & = {c \over a} \\ \\ \text{L.H.S} & = a (\alpha + 2)(\beta + 2) \\ & = a (\alpha \beta + 2\alpha + 2\beta + 4) \\ & = a [ \alpha \beta + 2(\alpha + \beta) + 4 ] \\ & = a \left[ {c \over a} + 2 \left(-{b \over a}\right) + 4 \right] \\ & = a \left( {c \over a} - {2b \over a} + 4 \right) \\ & = c - 2b + 4a \\ & = 4a - 2b + c \\ & = \text{R.H.S} \end{align}

(ii)

\begin{align} \text{Sum of roots} & = (\alpha + 2) + (\beta + 2) \\ & = \alpha + \beta + 4 \\ & = -{b \over a} + 4 \\ & = -{b \over a} + {4a \over a} \\ & = {-b + 4a \over a} \\ \\ \text{Product of roots} & = (\alpha + 2)(\beta + 2) \\ \\ \text{From part (i), } \phantom{0} a (\alpha + 2)(\beta + 2) & = 4a - 2b + c \\ (\alpha + 2)(\beta + 2) & = {4a - 2b + c \over a} \\ \\ \therefore \text{Product of roots} & = {4a - 2b + c \over a} \\ \\ \\ x^2 - \text{(Sum of roots)}x & + \text{(Product of roots)} = 0 \\ x^2 - \left(-b + 4a \over a\right)x & + {4a - 2b + c \over a} = 0 \\ ax^2 - (-b + 4a)x & + (4a - 2b + c) = 0 \\ ax^2 + (b - 4a)x & + (4a - 2b + c) = 0 \end{align}

\begin{align} 4x^2 + kx = 1 \\ 4x^2 + kx - 1 & = 0 \\ \\ [a = 4, b & = k, c = -1] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{k \over 4} \\ & = -{k \over 4} \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha)(\beta) & = {-1 \over 4} \\ \alpha \beta & = -{1 \over 4} \end{align}

(i)

\begin{align} \text{Sum of roots} & = (2\alpha + 3\beta) + (3\alpha + 2\beta) \\ & = 5\alpha + 5\beta \\ & = 5(\alpha + \beta) \\ & = 5\left(-{k \over 4}\right) \\ & = -{5k \over 4} \\ \\ \text{Product of roots} & = (2\alpha + 3\beta)(3\alpha + 2\beta) \\ & = 6\alpha^2 + 4\alpha \beta + 9\alpha \beta + 6\beta^2 \\ & = 6\alpha^2 + 6\beta^2 + 13 \alpha \beta \\ & = 6(\alpha^2 + \beta^2) + 13 \alpha \beta \\ & = 6[ (\alpha + \beta)^2 - 2\alpha \beta] + 13 \alpha \beta \\ & = 6(\alpha + \beta)^2 - 12\alpha \beta + 13 \alpha \beta \\ & = 6(\alpha + \beta)^2 + \alpha \beta \\ & = 6\left(-{k \over 4}\right)^2 + \left(-{1 \over 4}\right) \\ & = 6\left(k^2 \over 16\right) - {1 \over 4} \\ & = {3k^2 \over 8} - {2 \over 8} \\ & = {3k^2 - 2 \over 8} \\ \\ \\ x^2 - \text{(Sum of roots)}x & + \text{(Product of roots)} = 0 \\ x^2 - \left(-{5k \over 4}\right)x & + {3k^2 - 2 \over 8} = 0 \\ x^2 + {5k \over 4}x & + {3k^2 - 2 \over 8} = 0 \\ 8x^2 + 10k x & + 3k^2 - 2 = 0 \end{align}

(ii)

\begin{align} \text{Sum of roots} & = {3 \over \alpha + 2} + {3 \over \beta + 2} \\ & = {3(\beta + 2) + 3(\alpha + 2) \over (\alpha + 2)(\beta + 2) } \\ & = {3 \beta + 6 + 3 \alpha + 6 \over \alpha \beta + 2 \alpha + 2 \beta + 4} \\ & = {3(\alpha + \beta) + 12 \over \alpha \beta + 2(\alpha + \beta) + 4} \\ & = {3 \left(-{k \over 4}\right) + 12 \over -{1 \over 4} + 2 \left(-{k \over 4}\right) + 4} \\ & = {-{3 \over 4}k + 12 \over {15 \over 4} - {1 \over 2}k} \times {4 \over 4} \\ & = {-3k + 48 \over 15 - 2k} \\ \\ \text{Product of roots} & = \left( {3 \over \alpha + 2} \right) \left( {3 \over \beta + 2} \right) \\ & = {9 \over (\alpha + 2)(\beta + 2)} \\ & = {9 \over \alpha \beta + 2 \alpha + 2 \beta + 4} \\ & = {9 \over \alpha \beta + 2(\alpha + \beta) + 4} \\ & = {9 \over -{1 \over 4} + 2 \left(-{k \over 4}\right) + 4} \\ & = {9 \over {15 \over 4} - {1 \over 2}k} \times {4 \over 4} \\ & = {36 \over 15 - 2k} \\ \\ x^2 - \text{(Sum of roots)}x & + \text{(Product of roots)} = 0 \\ x^2 - \left({-3k + 48 \over 15 - 2k}\right)x + {36 \over 15 - 2k} & = 0 \\ x^2 + {3k - 48 \over 15 - 2k} x + {36 \over 15 - 2k} & = 0 \\ (15 - 2k)x^2 + (3k - 48)x + 36 & = 0 \end{align}

(i)

\begin{align} 2x^2 + 4x + 5 & = 0 \\ \\ [a = 2, b & = 4, c = 5] \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{4 \over 2} \\ & = -2 \\ \\ \text{Product of roots} & = {c \over a} \\ (\alpha)(\beta) & = {5 \over 2} \\ \alpha \beta & = {5 \over 2} \end{align}

(ii)

\begin{align} \text{Sum of roots} & = {1 \over \alpha} + {1 \over \beta} \\ & = {\beta \over \alpha \beta} + {\alpha \over \alpha \beta} \\ & = {\beta + \alpha \over \alpha \beta} \\ & = {-2 \over {5 \over 2}} \\ & = -{4 \over 5} \\ \\ \text{Product of roots} & = \left(1 \over \alpha\right)\left(1 \over \beta\right) \\ & = {1 \over \alpha \beta} \\ & = {1 \over {5 \over 2}} \\ & = {2 \over 5} \\ \\ \\ x^2 - \text{(Sum of roots)}x & + \text{(Product of roots)} = 0 \\ x^2 - \left(-{4 \over 5}\right)x & + {2 \over 5} = 0 \\ x^2 + {4 \over 5}x & + {2 \over 5} = 0 \\ 5x^2 + 4x & + 2 = 0 \end{align}

(iii)

\begin{align} \text{Let } \alpha \text{ & } \beta \text{ denote the roots} & \text{ of the equation }ax^2 + bx + c = 0. \\ \\ \text{Sum of roots} & = -{b \over a} \\ \alpha + \beta & = -{b \over a} \\ \\ \text{Product of roots} & = {c \over a} \\ \alpha \beta & = {c \over a} \\ \\ \\ \text{For the new equation, } & \text{the roots are } {1 \over \alpha} \text{ and } {1 \over \beta} \\ \\ \text{Sum of roots} & = {1 \over \alpha} + {1 \over \beta} \\ & = {\beta \over \alpha \beta} + {\alpha \over \alpha \beta} \\ & = {\beta + \alpha \over \alpha \beta} \\ & = {- {b \over a} \over {c \over a} } \\ & = -{b \over a} \div {c \over a} \\ & = -{b \over a} \times {a \over c} \\ & = -{b \over c} \\ \\ \text{Product of roots} & = \left(1 \over \alpha\right)\left(1 \over \beta\right) \\ & = {1 \over \alpha \beta} \\ & = {1 \over {c \over a}} \\ & = 1 \div {c \over a} \\ & = 1 \times {a \over c} \\ & = {a \over c} \\ \\ \\ x^2 - \text{(Sum of roots)}x & + \text{(Product of roots)} = 0 \\ x^2 - \left(-{b \over c}\right)x & + {a \over c} = 0 \\ x^2 + {b \over c}x & + {a \over c} = 0 \\ cx^2 + bx & + a = 0 \end{align}

(i)

\begin{align} x(2 - x) & = 3 \\ \\ \text{When } & x = \alpha, \\ \alpha(2 - \alpha) & = 3 \\ 2 \alpha - \alpha^2 & = 3 \\ - \alpha^2 & = 3 - 2 \alpha \\ \alpha^2 & = 2 \alpha - 3 \phantom{00} \text{--- (1)} \\ \\ (1) & \times \alpha, \\ \alpha^3 & = 2 \alpha^2 - 3\alpha \\ \\ \text{Since } & \alpha^2 = 2 \alpha - 3, \\ \alpha^3 & = 2(2 \alpha - 3) - 3 \alpha \\ \alpha^3 & = 4 \alpha - 6 - 3\alpha \\ \alpha^3 & = \alpha - 6 \phantom{00} \text{(Shown)} \end{align}

(ii)

\begin{align} \text{From (i), } \alpha^3 & = \alpha - 6 \\ \\ \text{Since } & x = \alpha \text{ or } \beta, \\ \beta^3 & = \beta - 6 \\ \\ \\ \therefore \alpha^3 + \beta^3 & = \alpha - 6 + \beta - 6 \\ & = \underbrace{\alpha + \beta}_\text{Need to find this} - 12 \\ \\ \\ x(2 - x) & = 3 \\ 2x - x^2 & = 3 \\ 0 & = x^2 - 2x + 3 \\ \\ [a = 1, b & = -2, c = 3] \\ \\ \alpha + \beta & = -{b \over a} \\ & = -{-2 \over 1} \\ & = 2 \\ \\ \\ \therefore \alpha^3 + \beta^3 & = 2 - 12 \\ & = -10 \end{align}

\begin{align} 2x^2 - kx + k & = 2 \\ 2x^2 - kx + (k - 2) & = 0 \\ \\ \text{Let } \alpha \text{ and } \beta \text{ denote the } & \text{roots of the equation} \\ \\ \text{Sum of roots, } \alpha + \beta & = -{b \over a} \\ & = -{-k \over 2} \\ & = {k \over 2} \\ \\ \text{Product of roots, } \alpha \beta & = {c \over a} \\ & = {k-2 \over 2} \\ & = {k \over 2} - {2 \over 2} \\ & = {k \over 2} - 1 \\ \\ \text{If } \alpha \text{ and } \beta \text{ are negative, } \alpha + \beta & < 0 \\ {k \over 2} & < 0 \\ \implies {k \over 2} - 1 & < - 1 \\ \alpha \beta & < - 1 \\ \\ \text{Since the product of 2 nega} & \text{tive numbers is positive, } \\ \text{it is not posible for both roo} & \text{ts to be negative} \end{align}

\begin{align} x^2 + 4(c + 2) & = (c + 4)x \\ x^2 - (c + 4)x + 4(c + 2) & = 0 \\ \\ \text{Sum of roots, } a + b & = -{-(c + 4) \over 1} \\ & = c + 4 \\ \\ \text{Product of roots, } ab & = {4(c + 2) \over 1} \\ & = 4(c + 2) \\ \\ \\ (a + b)^2 & = a^2 + 2(a)(b) + b^2 \\ (c + 4)^2 & = a^2 + 2ab + b^2 \\ c^2 + 2(c)(4) + 4^2 & = a^2 + 2[4(c + 2)] + b^2 \\ c^2 + 8c + 16 & = a^2 + 8(c + 2) + b^2 \\ c^2 + 8c + 16 & = a^2 + 8c + 16 + b^2 \\ c^2 & = a^2 + b^2 \\ \\ \text{By the converse of } & \text{Pythagoras theorem,} \\ \text{triangle is a right-a} & \text{ngled triangle} \\ \\ \therefore \text{Largest } & \text{angle is } 90^\circ \end{align}