1.6.3 Quadratic Equations, SPM Paper 2 (Long Questions)

Question 9:
Solve the following quadratic equation:
4x (x + 4) = 9 + 16x

Solution:

\begin{array}{l} 4 x (x + 4) = 9 + 16 x \\ 4 x^{2} + 16 x = 9 + 16 x \\ 4 x^{2} - 9 = 0 \\ {(2 x)}^{2} - 3^{2} = 0 \\ (2 x + 3) (2 x - 3) = 0 \\ 2 x + 3 = 0 or 2 x - 3 = 0 \\ 2 x = - 3 2 x = 3 \\ x = - \frac{3}{2} x = \frac{3}{2} \end{array}

Question 10:
Solve the following quadratic equation:
(x + 2)² = 2x + 7

Solution:

\begin{array}{l} {(x + 2)}^{2} = 2 x + 7 \\ x^{2} + 4 x + 4 = 2 x + 7 \\ x^{2} - 2 x - 3 = 0 \\ (x + 1) (x - 3) = 0 \\ x + 1 = 0 or x - 3 = 0 \\ x = - 1 x = 3 \end{array}

Question 11:
Solve the following quadratic equation:

- \frac{2}{3 x - 5} = \frac{x}{3 x - 1}

Solution:

\begin{array}{l} - \frac{2}{3 x - 5} = \frac{x}{3 x - 1} \\ - 2 (3 x - 1) = x (3 x - 5) \\ - 6 x + 2 = 3 x^{2} - 5 x \\ 3 x^{2} - 5 x + 6 x - 2 = 0 \\ 3 x^{2} + x - 2 = 0 \\ (3 x - 2) (x + 1) = 0 \\ 3 x - 2 = 0 or x + 1 = 0 \\ 3 x = 2 x = - 1 \\ x = \frac{2}{3} x = - 1 \end{array}

Question 12:
Solve the following quadratic equation:

\frac{5 x + 3 x^{2}}{1 - 2 x} = 6

Solution:

\begin{array}{l} \frac{5 x + 3 x^{2}}{1 - 2 x} = 6 \\ 5 x + 3 x^{2} = 6 - 12 x \\ 3 x^{2} + 5 x + 12 x - 6 = 0 \\ 3 x^{2} + 17 x - 6 = 0 \\ (3 x - 1) (x + 6) = 0 \\ 3 x - 1 = 0 or x + 6 = 0 \\ 3 x = 1 x = - 6 \\ x = \frac{1}{3} x = - 6 \end{array}