5

1 2x(x – 3) = (2 – x)(x + 1) 2x 2 – 6x = 2x + 2 – x 2 – x 3x 2 – 7x – 2 = 0

x = – (–7) ± (–7)2 – 4(3)(–2)

2(3)

x = 7 ± 49 + 24

6

x = 7 ± 8.54406

x = 2.591 or –0.2573

2 The roots are 23

and – 52

.

Sum of roots = 23

+ – 52 = – 11

6

Product of roots = 23

× – 52 = – 5

3 The quadratic equation is

x2 + 116

x – 53

= 0

6x 2 + 11x – 10 = 0

x 2 – (sum of roots)x + (product of roots) = 0

3 y = 5x – 2 … 1 y = 3x 2 + 3x + k … 2 Substituting 2 into 1 : 3x2 + 3x + k = 5x – 2 3x2 – 2x + k + 2 = 0

a = 3, b = –2, c = k + 2

In the case where a curve does not meet a straight line, b2 – 4ac < 0 is applied.

b2 – 4ac < 0 (–2)2 – 4(3)(k + 2) < 0 4 – 12k – 24 < 0 –12k – 20 < 0 –12k < 20

k > 20–12

k > –123

4 7 – 2(1 + x)2 = x(x + 5) 7 – 2(1 + 2x + x 2) = x 2 + 5x 7 – 2 – 4x – 2x 2 = x 2 + 5x 3x 2 + 9x – 5 = 0

x = –b ± b2 – 4ac

2a

x = –9 ± 92 – 4(3)(–5)

2(3)

x = –9 ± 141

6 x = 0.4791 or –3.479

5 9x 2 + qx + 1 = 4x 9x 2 + qx – 4x + 1 = 0 9x 2 + (q – 4)x + 1 = 0 a = 9, b = q – 4, c = 1 If the equation has equal roots, then b2 – 4ac = 0 (q – 4)2 – 4(9)(1) = 0 q2 – 8q + 16 – 36 = 0 q2 – 8q – 20 = 0 (q + 2)(q – 10) = 0 q = –2 or 10

6 (a) mx 2 + nx + 12 = 0 If the quadratic equation has equal roots, b2 – 4ac = 0 n2 – 4m(12) = 0 n2 = 48m

m = n2

48

(b) If m = 2 and n = –11, 2x 2 – 11x + 12 = 0 (2x – 3)(x – 4) = 0

x = 32

or 4

7 x 2 – px – 6 = 0 When x = –1 (–1)2 – p(–1) – 6 = 0 1 + p – 6 = 0 p – 5 = 0 p = 5

Form 4: Chapter 2 (Quadratic Equations)SPM Practice

Fully Worked Solutions

Paper 1

6

8 Sum of roots = – ba

2 + m + 1 = –(n – 2) 3 + m = 2 – n m + n = –1 ... 1

Product of roots = ca

2(m + 1) = 6 m = 2 From 1 : 2 + n = –1 n = –3

9 x (x – 6) = h – 4k x2 – 6x + 4k – h = 0

a = 1, b = –6, c = 4k – h

For equal roots, b2 – 4ac = 0 (–6)2 – 4(1)(4k – h) = 0 36 – 16k + 4h = 0 9 – 4k + h = 0 h = 4k – 9

Paper 2

1 (a) x2 + 3(3x + k) = 0

x2 + 9x + 3k = 0

Sum of roots = – ba

p + 2p = –9 3p = –9 p = –3

Product of roots = ca

p(2p) = 3k 2p2 = 3k 2(–3)2 = 3k 18 = 3k k = 6

(b) The new roots are p – 2 = –3 – 2 = –5 and p + 4 = –3 + 4 = 1. Sum of roots = –5 + 1 = –4 Product of roots = –5(1) = –5 The quadratic equation is x2 + 4x – 5 = 0.