# How do I solve a quadratic equation by factorising?

A quadratic equation is one that includes x^{2 }as the highest power of x. Factorising is achieved in 3 steps. Let’s consider the example x^{2}-3x-3=1

*1) Put the equation into the form ax ^{2}+bx+c=0*

x^{2}-3x-4=0

2*) Factorise*

We need two numbers that

- add together to get -3

- Multiply together to get -4

-4x1=-4 and -4+1=-3

Thus, factorising gives (x-4)(x+1)=0

*3) Solve the equation!*

If two numbers are multiplied together to give 0, one of them must be 0. Thus:

x-4=0 and x=4

x+1=0 and x=-1

The equation has been solved

**Additional points:**

- This technique can be applied to finding the points of intersection on the x axis for a quadratic graph. For example, y=x^{2}-3x-4. At the x axis, y=0 so you can work out x as above.

- Harder quadratic equations can also be solved by factorising. For example when a isn't 1.

2x^{2} + 7x + 3=0

Find two numbers that multiply to give 2x3 (6) and add to give 7. In this case, 6 and 1.

Split 7x into 6x +x

2x^{2} + 6x+x + 3=0

Factorise each part by taking out a common factor.

2x(x+3)+1(x + 3)=0

The sames as

(2x+1)(x+3)=0

thus x = -1/2 or x=-3

__Practice questions__

1. Solve by factorising

*x*^{2} + 6*x* + 8=0

*x*^{2} – 8*x* + 16 = 0

2. Find the points of intersection with the x axis for

y=*x*^{2} – 14*x* + 48

and sketch this function