## Solving Quadratic Equations by Factorising

Solutions for equations are found when an expression evaluates to zero. One method for solving a quadratic equation is by factorising. After factorising, find the value of x that makes each bracket zero.

Rearrange the equation so that one side of the equation is equal to zero. Next, factorise the expression. Then find the two values of x that will solve the equation. In some instances the two values may be the same - x^2+6x+9 factorises to (x+3)(x+3) which gives you solutions of -3 and -3.

Note that not all quadratic expressions have a solution: for example, x^2-3x+7 cannot be solved except by using advanced mathematics.

## Example 1

Solve x^2-4x-5=0

x^2-4x-5 = 0

Factorise the expression:

(x+1)(x-5) = 0

Determine what values of x causes each set of brackets to equal zero:

(x+1)=0 when x=-1, and (x-5) =0 when x=5

The solutions are x=-1 text( and ) x=5

Check:

(-1)2 - 4(-1) - 5 =0✔

(502 - 4(5) - 5 = 0✔

Answer: x=-1 text( and ) x=5

## Example 2

Solve 2x^2-7x=4

2x^2-7x=4

Rearrange the equation so that one side of the equation equals zero:

2x^2 - 7x - 4 = 0

Factorise the equation:

(2x+1)(x-4)=0

What values of x will make each bracket zero?

(2x+1)=0 when x=-0.5

and (x-4) =0 when x=4

The solutions are therefore x=-0.5 text( and ) x=4

Check:

2(-0.5)2 - 7(-0.5) = 4✔

2(4)2 - 7(4) = 4✔

Answer: x=-frac(1)(2) text( and ) x=4