By Joe Gill

Introduction

There are several ways of solving Quadratic equations. These are

FACTORISING

COMPLETING THE SQUARE

TRIAL & IMPROVEMENT

Here is an example of how to solve a quadratic equation using factorising:-

Factorising a quadratic means putting the equation into two brackets. The standard format for quadratics is ax² + bx +c = 0.

Solve x2-x+12=0 by factorising

1. Firstly rearrange it x²-x=12

2. a=1 so the initial brackets are :- (x    ) (x   )

3. We now want to look at all the pairs of numbers that multiply to give c (=12), but which also add or subtract to give the value of b:

1 x 12 gives: 13 or 11

2 x 6   give: 8 or 4

3 x 4    give 7 or 1 (this is the value of b)

4. So 3 & 4 will give b + or – 1, so put them in the brackets: - (x  3) (x   4)= 0

5. Now fill in the +/- signs so that the 3 &4 add/subtract to give -1 (=b),so we must have +3 and -4 so we’ll have (x+3) (x-4)

6. Now check by expanding the brackets out and see of they give x²-x = 12.

7. Now we have to work out the roots. A simple way to work out the roots is to just switch the +/- signs on the two numbers in the bracket i.e. x=-3 or +4

Examples of some quadratic equations Solved by Factorising

Here are a few examples of factorising & solving equations where a=1:-

1.)    X² +7x+10= 0

(x   ) (x    )

5+2=7 &5x2=10

(x  5)(x   2)

(x+5)(x+2)

x= -5 or -2

2.)    x² + 6x -40=0

(x+10)(x-4)

x= -10 or +4

3.)     x² + 10x + 16 = 0

(x+2)(x+8)

x = -2 or -8

4.)     x² + 12x + 27= 0

(x+9)(x+3)

x= -9 or – 3

5.)     x²+14x +40 = 0

(x+4)(x+10)

x=-4 or-10

6.)     x²+ 15x + 36 = 0

(x+12)(x+3)

x= -12 or – 3

7.)     x²-3x-18=0

(x+3)(x-6)

x= -3 or + 6

8.)     x² -7x -30=0

(x+3)(x-10)

x=-3 or +10

9.)     x²+ 4x – 32=0

(x+4) (x-8)

x= -4 or +8

A Table of Results for Quadratic Equations

Here is a table to summarise the ...