Factorising Quadratics (OCR GCSE Maths)

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Jamie W

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Jamie W

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Factorising Simple Quadratics

What is a quadratic expression?

  • A quadratic expression is in the form:
    • ax2 + bx + c (as long as a ≠ 0)
  • If there are any higher powers of x (like x3 say) then it is not a quadratic
  • If a = 1 e.g. x squared minus 2 x minus 8, it can be called a “monic” quadratic expression
  • If a ≠ 1 e.g. 2 x squared minus 2 x minus 8, it can be called a “non-monic” quadratic expression

 

Method 1: Factorising "by inspection"

  • This is shown easiest through an example; factorising x squared minus 2 x minus 8
  • We need a pair of numbers that for x squared plus b x plus c
    • multiply to c
      • which in this case is -8
    • and add to b
      • which in this case is -2
    • -4 and +2 satisfy these conditions
    • Write these numbers in a pair of brackets like this: 
      • open parentheses x plus 2 close parentheses open parentheses x minus 4 close parentheses

 

Method 2: Factorising "by grouping"

  • This is shown easiest through an example; factorising x squared minus 2 x minus 8
  • We need a pair of numbers that for x squared plus b x plus c
    • multiply to c
      • which in this case is -8
    • and add to b
      • which in this case is -2
    • 2 and -4 satisfy these conditions
    • Rewrite the middle term by using 2x and -4x
      • x squared plus 2 x minus 4 x minus 8
    • Group and factorise the first two terms, using x as the highest common factor, and group and factorise the second two terms, using -4 as the factor
      • x open parentheses x plus 2 close parentheses minus 4 open parentheses x plus 2 close parentheses
    • Note that these now have a common factor of (x + 2) so this whole bracket can be factorised out
      • open parentheses x plus 2 close parentheses open parentheses x minus 4 close parentheses

 

Method 3: Factorising "by using a grid"

  • This is shown easiest through an example; factorising x squared minus 2 x minus 8
  • We need a pair of numbers that for x squared plus b x plus c
    • multiply to c
      • which in this case is -8
    • and add to b
      • which in this case is -2
    • -4 and +2 satisfy these conditions
    • Write the quadratic equation in a grid (as if you had used a grid to expand the brackets), splitting the middle term as -4x and 2x
    • The grid works by multiplying the row and column headings, to give a product in the boxes in the middle
     
  x2 -4x
  +2x -8

  • Write a heading for the first row, using x as the highest common factor of x2 and -4x
     
x x2 -4x
  +2x -8

  • You can then use this to find the headings for the columns, e.g. “What does x need to be multiplied by to give x2?”
  x -4
x x2 -4x
  +2x -8

  • We can then fill in the remaining row heading using the same idea, e.g. “What does x need to be multiplied by to give +2x?”
  x -4
x x2 -4x
+2 +2x -8

  • We can now read-off the factors from the column and row headings
    • open parentheses x plus 2 close parentheses open parentheses x minus 4 close parentheses

  

Which method should I use for factorising simple quadratics?

  • The first method, by inspection, is by far the quickest so is recommended in an exam for simple quadratics (where a = 1)
  • However the other two methods (grouping, or using a grid) can be used for harder quadratic equations where ≠ 1 so you should learn at least one of them too

Examiner Tip

  • As a check, expand your answer and make sure you get the same expression as the one you were trying to factorise.

Worked example

(a) Factorise x squared minus 4 x minus 21.

We will factorise by inspection.

We need two numbers that:

multiply to -21, and sum to -4

-7, and +3 satisfy this

Write down the brackets.

 (x + 3)(x - 7)

  

(b) Factorise x squared minus 5 x plus 6.

We will factorise by splitting the middle term and grouping.

We need two numbers that:

multiply to 6, and sum to -5

-3, and -2 satisfy this

Split the middle term.

x2 - 2x - 3x + 6

Factorise x out of the first two terms.

x(x - 2) - 3x +6

Factorise -3 out of the last two terms.

x(x - 2) - 3(x - 2)

These have a common factor of (x - 2) which can be factored out.

(x - 2)(x - 3)

 

(c) Factorise x squared minus 2 x minus 24.

We will factorise by using a grid.

We need two numbers that:

multiply to -24, and sum to -2

+4, and -6 satisfy this

Use these to split the -2x term and write in a grid.

     
  x2 +4x
  -6x -24

 
Write a heading using a common factor for the first row:

     
x x2 +4x
  -6x -24

 
Work out the headings for the rows, e.g. “What does x
need to be multiplied by to make x2?”

  x +4
x x2 +4x
  -6x -24

 
Repeat for the heading for the remaining row, e.g. “What does x
need to be multiplied by to make -6x?”

  x +4
x x2 +4x
-6 -6x -24

 
Read-off the factors from the column and row headings.

(x + 4)(x - 6)

 

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Factorising Harder Quadratics

How do I factorise a harder quadratic expression?

Factorising a ≠ 1 "by grouping"

  • This is shown easiest through an example; factorising 4 x squared minus 25 x minus 21
  • We need a pair of numbers that for a x squared plus b x plus c
    • multiply to ac
      • which in this case is 4 × -21 = -84
    • and add to b
      • which in this case is -25
    • -28 and +3 satisfy these conditions
    • Rewrite the middle term using -28x and +3x
      • 4 x squared minus 28 x plus 3 x minus 21
    • Group and factorise the first two terms, using 4x as the highest common factor, and group and factorise the second two terms, using 3 as the factor
      • 4 x open parentheses x minus 7 close parentheses plus 3 open parentheses x minus 7 close parentheses
    • Note that these terms now have a common factor of (x - 7) so this whole bracket can be factorised out, leaving 4x + 3 in its own bracket
      • open parentheses x minus 7 close parentheses open parentheses 4 x plus 3 close parentheses

 

Factorising a ≠ 1 "by using a grid"

  • This is shown easiest through an example; factorising 4 x squared minus 25 x minus 21
  • We need a pair of numbers that for a x squared plus b x plus c
    • multiply to ac
      • which in this case is 4 × -21 = -84
    • and add to b
      • which in this case is -25
    • -28 and +3 satisfy these conditions
    • Write the quadratic equation in a grid (as if you had used a grid to expand the brackets), splitting the middle term as -28x and +3x
    • The grid works by multiplying the row and column headings, to give a product in the boxes in the middle
     
  4x2 -28x
  +3x -21

    • Write a heading for the first row, using 4x as the highest common factor of 4x2 and -28x
     
4x 4x2 -28x
  +3x -21

    • You can then use this to find the headings for the columns, e.g. “What does 4x need to be multiplied by to give 4x2?”
  x -7
4x 4x2 -28x
  +3x -21

 

    • We can then fill in the remaining row heading using the same idea, e.g. “What does x need to be multiplied by to give +3x?”
  x -7
4x 4x2 -28x
+3 +3x -21

    • We can now read-off the factors from the column and row headings
      • open parentheses x minus 7 close parentheses open parentheses 4 x plus 3 close parentheses

Examiner Tip

As a check, expand your answer and make sure you get the same expression as the one you were trying to factorise.

Worked example

(a) Factorise 6 x squared minus 7 x minus 3.

 
We will factorise by splitting the middle term and grouping.

We need two numbers that:

multiply to 6 × -3 = -18, and sum to -7

-9, and +2 satisfy this

Split the middle term.

6x2 + 2x - 9x - 3

Factorise 2x out of the first two terms.

2x(3x + 1) - 9x - 3

Factorise -3 of out the last two terms.

2x(3x + 1) - 3(3x + 1)

These have a common factor of (3x + 1) which can be factored out.

(3x + 1)(2x - 3)

 

(b) Factorise 10 x squared plus 9 x minus 7.

 
We will factorise by using a grid.

We need two numbers that:

multiply to 10 × -7 = -70, and sum to +9

-5, and +14 satisfy this

Use these to split the 9x term and write in a grid.

     
  10x2 -5x
  +14x -7


Write a heading using a common factor for the first row:

     
5x 10x2 -5x
  +14x -7

Work out the headings for the rows, e.g. “What does 5x need to be multiplied by to make 10x2?”

  2x -1
5x 10x2 -5x
  +14x -7


Repeat for the heading for the remaining row, e.g. “What does 2x
need to be multiplied by to make +14x?”

  2x -1
5x 10x2 -5x
+7 +14x -7


Read-off the factors from the column and row headings.

(2x - 1)(5x + 7)

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Difference Of Two Squares

What is the difference of two squares?

  • When a "squared" quantity is subtracted from another "squared" quantity, you get the difference of two squares
    • for example,
      • a2 - b2
      • 92 - 52
      • (x + 1)2 - (x - 4)2
      • 4m2 - 25n2, which is (2m)2 - (5n)2

 

How do I factorise the difference of two squares?

  • Expand the brackets (a + b)(a - b)
    • = a2 - ab + ba - b2
    • ab is the same quantity as ba, so -ab and +ba cancel out
    • = a2 - b2
  • From the working above, the difference of two squares, a2 - b2, factorises to

open parentheses a plus b close parentheses open parentheses a minus b close parentheses

  • It is fine to write the second bracket first, (a - b)(a + b)
    • but the a and the b cannot swap positions
      • a2 - b2 must have the a's first in the brackets and the b's second in the brackets

Examiner Tip

  • The difference of two squares is a very important rule to learn as it often appears in harder questions involving factorisation, e.g. in algebraic fractions
  • The word difference in maths means a subtraction, it should remind you that you are subtracting one squared term from another
  • You should be able to recognise factorised difference of two squares expressions

Worked example

(a)
Factorise  9 x squared minus 16.

 

Recognise that 9 x squared and 16 are both squared terms and the second term is subtracted from the first term - you can factorise using the difference of two squares.

9 x squared minus 16 equals open parentheses 3 x close parentheses squared minus open parentheses 4 close parentheses squared

Rewrite the expression with the square root of each term added together in the first bracket and subtracted from each other in the second bracket.

 

(b)
Factorise 4 x squared minus 25.

Recognise that 4 x squared and 25 are both squared terms and the second term is subtracted from the first term - you can factorise using the difference of two squares.

4 x squared minus 25 equals open parentheses 2 x close parentheses squared minus open parentheses 5 close parentheses squared

Rewrite the expression with the square root of each term added together in the first bracket and subtracted from each other in the second bracket.

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Jamie W

Author: Jamie W

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.