How To Factorise Simple Quadratic Expressions

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Subject: Maths » GCSE Maths
Last updated: 13/03/2018
Tags: algebra, factorisation, gcse maths, how to, quadratic equations

Quadratic expressions follow the pattern,   Ax2 + Bx + C.

We shall be looking at simple quadratic expressions, namely where A = 1, which will follow the pattern,  x2 + Bx + C. When this expression is factorised, the result will be the product of two brackets.

x2 + Bx + C = ( x     ) ( x     ) 

Signs
First we need to work out what the signs in the brackets will be. We look at the constant C, if it is POSITIVE, then both the signs in the brackets will be the SAME. We then look at B, if it is POSITIVE then both the signs in the brackets will be POSITIVE, if it is NEGATIVE both the signs in the brackets will be NEGATIVE.    

x2 + Bx + C = ( x  +   ) ( x  +   )
x2 - Bx + C = ( x -   ) ( x  -   )

But if C is NEGATIVE then the signs in the brackets will be DIFFERENT.

x2 + Bx - C = ( x  +   ) ( x  -   )  or  ( x  -   ) ( x  +   )
x2 - Bx - C = ( x  +   ) ( x  -   )  or  ( x  -   ) ( x  +   ) 

Numbers
Now to work out the numbers that go into the brackets. If C is POSITIVE then the missing numbers are the two FACTORS of C that ADD together to make B.

e.g. x2 + 5x + 6
C is positive, so both the signs will be the same. B is positive so both the signs will be positive
So x2 + 5x + 6 = ( x  +   ) ( x  +   )
The two factors of 6 that add up to 5 are 3 and 2
So x2 + 5x + 6 = ( x  + 3 ) ( x  + 2 )

e.g. x2 - 5x + 4
C is positive, so both the signs will be the same. B is negative so both the signs will be negative
So x2 - 5x + 4 = ( x  -   ) ( x  -   )
The two factors of 4 that add up to 5 are 4 and 1
So x2 - 5x + 4 = ( x  - 4 ) ( x  - 1 )

Have a go at these:
x2 + 7x + 6     x2 - 10x + 21      x2 - 13x + 12          x2 + 9x + 20

If C is NEGATIVE then the missing numbers are the two FACTORS of C that have a DIFFERENCE of B. ALSO if B is POSITIVE then the larger number goes with the POSITIVE sign, if B is NEGATIVE then the larger number goes with the NEGATIVE sign

e.g. x2 + 4x - 12
C is negative, so both the signs will be different.
So x2 + 4x - 12 = ( x  +   ) ( x  -   )
The two factors of 12 that have a difference of 4 are 6 and 2. B is positive so the 6 goes with the positive sign
So x2 + 4x - 12 = ( x  + 6 ) (x  - 2 )
Compared with x2 - 4x - 12 = ( x  + 2 ) ( x  -  6 )

Have a go at these:
x2 + x - 6        x2 - 5x - 24        x2 + 8x - 9         x2 - 5x - 14

Answers:
x2 + 7x + 6 = ( x  + 1 ) ( x  + 6 )
x2 - 10x + 21 = ( x  - 3 ) ( x  - 7 )
x2 - 13x + 12 = ( x - 1 ) ( x  - 12 )
x2 + 9x + 20 = ( x  + 4 ) ( x  + 5 )
x2 + x - 6 = ( x  + 6 ) ( x  - 1 )
x2 - 5x - 24 = ( x  - 8 ) ( x  + 3 )
x2 +8 x - 9 = ( x  + 9 ) ( x  - 1 )
x2 + 5x - 14 = ( x  - 7 ) ( x  + 2 )


Tril More 11+ exam Tutor (Birmingham)

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