# Solving Quadratic Equations by Factorisation. By I Porter

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Solving Quadratic Equations by Factorisation. By I Porter

Introduction An equation which can be written in the form ax2 + bx + c = 0 , where a ≠0, is called a quadratic equation. Some examples of quadratic equations are: x2 - 4 = 0, x2 + 5x = 0, x2 + 6x + 8 = 0, 3x2 + 2x - 5 = 0 and so on. Quadratic equation can be written in many different forms, such as: x2 = 16, x2 = 8x, x2 = 4x +12. In these case the quadratic equation must be rearranged so that one side is ZERO (0). i.e. x2 = 16 becomes x = 0. Quadratic equations can be solved by one of 3 methods: * Factorisation * Complete the Square Method * Quadratic Formula

Factorisation Method We can solve quadratic equations using a basic law of numbers called the null factor law: If a x b = 0, either a = 0 or b = 0 or both a = 0 and b = 0. If a quadratic is written in the form of two factors and their product is zero, either one of the factors or both of the factors must equal zero (0). For example, if we have (x + a) (x + b) = 0 , either x + a = 0 or x + b = 0 or both (x + a) = 0 and (x + b) = 0.

Examples: Factorise and solve each of the following.
a) x2- 5x = 0 Factorise fully! b) x = 0 Factorise fully! (x - 5) x = 0 (x - 4)(x + 4) = 0 So, either So, either x - 5 = 0 or x = 0 x - 4 = 0 or x + 4 = 0 x = 5 x = 4 x = -4 The solutions are x = 0 and x = 5. The solutions are x = -4 and x = 4.

Examples c) x2 - 12 = 0 d) x2 + 4x - 12 = 0 (x + 6)(x - 2) = 0
Factorise fully! Factorise fully! (x + 6)(x - 2) = 0 So, either So, either x + 6 = 0 or x - 2 = 0 x = -6 x = 2 The solutions are and The solutions are x = -6 and x = 2.

Examples f) 2x2 = 19x - 24 e) 3x = 4 - x2 2x2 - 19x + 24 = 0
Rearrange, RHS = 0 e) 3x = 4 - x2 Rearrange, RHS = 0 Factorise fully! Sum/product 2x2 - 19x + 24 = 0 x2 + 3x - 4 = 0 Factorise fully! 2 x 24 = -1 x -48 sum = -49 = -2 x -24 sum -26 = -3 x -16 sum = -19 2x2 - 3x - 16x+ 24 = 0 (x - 1)(x + 4) = 0 So, either x(2x - 3) - 8(2x - 3) = 0 x - 1 = or x + 4 = 0 (x - 8)(2x - 3) = 0 So, either x = 1 x = -4 x - 8 = 0 or 2x - 3 = 0 The solutions are x = -4 and x = 1. x = 8 x = 3/2 The solutions are x = 8 and x = 3/2.

Exercise: Solve each of the following.
1) x2 + 6x = 0 2) x = 0 3) x2 - 6x = 16 Sol: x = 0, x = -6 Sol: x = -5, x = 5 Sol: x = -2, x = 8 4) 3x = 0 5) x2 - 12x + 32 = 0 6) 2x2 + x = 3 Sol: x = -3, x = 3 Sol: x = 4, x = 8 Sol: x = -1, x = 3/2 7) 4x2 + 11x + 6 = 0 8) 5x2 = 6x - 1 9) 5x - 3x2 = -28 Sol: x = -2, x = -3/4 Sol: x = 1, x = 1/5 Sol: x = 4, x = -7/3