Solving quadratic equations

Factorisation

Type 1: No constant termSolve x2 – 6x = 0

x (x – 6) = 0

x = 0 or x – 6 = 0

Solutions: x = 0 or x = 6

Graph of y = x2 – 6x

y

x

x = 3

(0, 0) (6, 0)

y = x2 – 6x

Vertex(3, -9)

Solving quadratic equations

Factorisation

Type 2: No linear term

Solve x2 – 9 = 0

x2 – 9 = x2 – 32 (x + 3)(x – 3) = 0

Solutions: x = -3 or x = 3

Graph of y = x2 – 9

y

x

x = 0

(-3, 0) (3, 0)

y = x2 – 9

Vertex(0, -9)

Difference of two squares

= (x + 3)(x – 3)

x + 3 = 0 or x – 3 = 0

Solving quadratic equations

Factorisation

Type 3: All three terms

Solve x2 – x – 12 = 0

x2 – x – 12 =

a b = -12 and a + b = -1

Solutions: x = -3 or x = 4

Graph of y = x2 – x - 12

y

x

x = ½

(-3, 0) (4, 0)

y = x2 – x -12

Vertex( ½ , -12 ¼ )

(x )(x )

(x + 3)(x – 4) = 0

a = 3 and b = -4

x + 3 = 0 or x – 4 = 0

Factorise the following quadratic equations hence solve them.

1. 3x2 – 2x = 0

x(3x – 2) = 0 x = 0 or 3x – 2 = 0 x = 0 or x = 2/3

2. 4x2 – 5 = 0

(2x + 5)(2x - 5) = 0

2x + 5 = 0 or

Difference of two squares

2x - 5 = 0

2

5x

2

5xor

3. x2 + 2x = 8

x2 + 2x – 8 = 0(x + 4)(x – 2) = 0 x + 4 = 0 or x – 2 = 0

4. 5x2 + 13x - 6 = 0

(5x - 2)(x + 3) = 0

5x – 2 = 0 or x + 3 = 0

x = - 4 or x = 2

x = 2/5 or x = - 3