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Factorising Quadratics (CIE IGCSE Maths: Extended)

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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 ā‰  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 ā€œmonicā€ quadratic expression
  • If a ā‰  1 e.g.Ā 2 x squared minus 2 x minus 8, it can be called a ā€œnon-monicā€ quadratic expression

How to Factorise Quadratics

Method 1: Factorising "by inspection"

  • This is shown easiest through an example; factorisingĀ x squared minus 2 x minus 8
  • We need a pair of numbers that forĀ 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:Ā 
      • open parentheses x plus 2 close parentheses open parentheses x minus 4 close parentheses

Ā 

Method 2: Factorising "by grouping"

  • This is shown easiest through an example; factorisingĀ x squared minus 2 x minus 8
  • We need a pair of numbers that forĀ 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

Ā 

Method 3: FactorisingĀ "by using a grid"

  • This is shown easiest through an example; factorisingĀ x squared minus 2 x minus 8
  • We need a pair of numbers that forĀ 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
Ā  Ā  Ā 
Ā  x2 -4x
Ā  +2x -8

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

  • You can then use this to find the headings for the columns, e.g. ā€œWhat does x need to be multiplied by to give x2?ā€
Ā  x -4
x x2 -4x
Ā  +2x -8

  • We can then fill in the remaining row heading using the same idea, e.g. ā€œWhat does x need to be multiplied by to give +2x?ā€
Ā  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

Ā Ā 

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 aĀ ā‰  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) FactoriseĀ 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.

Ā (x + 3)(x - 7)

Ā Ā 

(b) FactoriseĀ 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)

Ā 

(c) FactoriseĀ 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.

Ā  Ā  Ā 
Ā  x2 +4x
Ā  -6x -24

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

Ā  Ā  Ā 
x x2 +4x
Ā  -6x -24

Ā 
Work out the headings for the rows, e.g. ā€œWhat does x
need to be multiplied by to make x2?ā€

Ā  x +4
x x2 +4x
Ā  -6x -24

Ā 
Repeat for the heading for the remaining row, e.g. ā€œWhat does x
need to be multiplied by to make -6x?ā€

Ā  x +4
x x2 +4x
-6 -6x -24

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

(x + 4)(x - 6)

Ā 

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Factorising Harder Quadratics

How do I factorise a harder quadratic expression?

Factorising a ā‰  1 "by grouping"

  • This is shown easiest through an example; factorisingĀ 4 x squared minus 25 x minus 21
  • We need a pair of numbers that forĀ a x squared plus b x plus c
    • multiply to ac
      • which in this case is 4 Ɨ -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

Ā 

Factorising a ā‰  1Ā "by using a grid"

  • This is shown easiest through an example; factorisingĀ 4 x squared minus 25 x minus 21
  • We need a pair of numbers that forĀ a x squared plus b x plus c
    • multiply to ac
      • which in this case is 4 Ɨ -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
Ā  Ā  Ā 
Ā  4x2 -28x
Ā  +3x -21

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

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

Ā 

    • We can then fill in the remaining row heading using the same idea, e.g. ā€œWhat does x need to be multiplied by to give +3x?ā€
Ā  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) FactoriseĀ 6 x squared minus 7 x minus 3.

Ā 
We will factorise by splitting the middle term and grouping.

We need two numbers that:

multiply to 6 Ɨ -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)

Ā 

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

Ā 
We will factorise by using a grid.

We need two numbers that:

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

-5, and +14 satisfy this

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

Ā  Ā  Ā 
Ā  10x2 -5x
Ā  +14x -7


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

Ā  Ā  Ā 
5x 10x2 -5x
Ā  +14x -7

Work out the headings for the rows, e.g. ā€œWhat does 5x need to be multiplied by to make 10x2?ā€

Ā  2x -1
5x 10x2 -5x
Ā  +14x -7


Repeat for the heading for the remaining row, e.g. ā€œWhat does 2x
need to be multiplied by to make +14x?ā€

Ā  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

Ā 

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'sĀ 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 toĀ recognise factorised difference of two squares expressions

Worked example

(a)
FactoriseĀ Ā 9 x squared minus 16.

Ā 

Recognise that 9 x squared andĀ 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.

stretchy left parenthesis 3 x plus 4 stretchy right parenthesis stretchy left parenthesis 3 x minus 4 stretchy right parenthesis

Ā 

(b)
FactoriseĀ 4 x squared minus 25.

Recognise that 4 x squared andĀ 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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Quadratics Factorising Methods

How do I know if it factorises?

  • Method 1: Use a calculator to solve the quadratic expression equal to 0
    • If the solutions are integers or fractions (without square roots), then the quadratic expression factorises
  • Method 2: Find the value under the square root in the quadratic formula, b2 ā€“ 4ac (called the discriminant)
    • If this number is a perfect square number, then the quadratic expression factorises

Ā 

Which factorisation method should I use for a quadratic expression?

  • Does it have 2 terms only?
    • Yes, likeĀ x squared minus 7 x
      • Use "basic factorisation" to take out the highest common factor
      • x open parentheses x minus 7 close parentheses
    • Yes, likeĀ x squared minus 9
      • Use the "difference of two squares" to factorise
      • open parentheses x plus 3 close parentheses open parentheses x minus 3 close parentheses
  • Does it have 3 terms?
    • Yes, starting with x2Ā like x squared minus 3 x minus 10
      • Use "factorising simple quadratics" by finding two numbers that add to -3 and multiply to -10
      • open parentheses x plus 2 close parentheses open parentheses x minus 5 close parentheses
    • Yes, starting with ax2 like 3 x squared plus 15 x plus 18
      • Check to see if the 3 in front of x2 is a common factor for all three terms (which it is in this case), then use "basic factorisation" to factorise it outĀ first
      • 3 open parentheses x squared plus 5 x plus 6 close parentheses
      • The quadratic expression inside the brackets is now x2 +... , which factorises more easily
      • 3 open parentheses x plus 2 close parentheses open parentheses x plus 3 close parentheses
    • Yes, starting with ax2 like 3 x squared minus 5 x minus 2
      • The 3 in front of x2 is not a common factor for all three term
      • Use "factorising harder quadratics", for example factorising by grouping or factorising using a grid
      • open parentheses 3 x plus 1 close parentheses open parentheses x minus 2 close parentheses

Worked example

FactoriseĀ  negative 8 x squared plus 100 x minus 48.

Ā 
Spot the common factor of -4 and put outside a set of brackets, work out the terms inside the brackets by dividing the terms in the original expression by -4.

negative 8 x squared plus 100 x minus 48 equals negative 4 open parentheses 2 x squared minus 25 x plus 12 close parentheses

Check the discriminant for the expression inside the brackets, open parentheses b squared minus 4 a c close parentheses, to see if it will factorise.

table row blank blank cell open parentheses negative 25 close parentheses squared minus 4 cross times 2 cross times 12 end cell row blank equals cell 625 minus 96 end cell row blank equals 529 end table

529 equals 13 squared, it is a perfect square so the expression will factorise.

Proceed with factorising 2 x squared minus 25 x plus 12 as you would for a harder quadratic, where a not equal to 1.
"+12" means the signs will be the same.
"-25" means that both signs will be negative.

a cross times c equals 2 cross times 12 equals 24

The only numbers which multiply to give 24 and follow the rules for the signs above are:
open parentheses negative 1 close parentheses cross times open parentheses negative 24 close parentheses andĀ open parentheses negative 2 close parentheses cross times open parentheses negative 12 close parenthesesand open parentheses negative 3 close parentheses cross times open parentheses negative 8 close parentheses andĀ open parentheses negative 4 close parentheses cross times open parentheses negative 6 close parentheses
but only the first pair add to giveĀ negative 25.

Split theĀ negative 25 x term intoĀ negative 24 x minus x.

table row blank blank cell 2 x squared minus 24 x minus x plus 12 end cell end table

Group and factorise the first two terms, usingĀ 2 x as the highest common factor and group and factorise the last two terms using 1 as the highest common factor.

table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 2 x open parentheses x minus 12 close parentheses plus 1 open parentheses x minus 12 close parentheses end cell end table

These factorised terms now have a common term of open parentheses x minus 12 close parentheses, so this can now be factorised out.

open parentheses 2 x plus 1 close parentheses open parentheses x minus 12 close parentheses

Put it all together.

negative 8 x squared plus 100 x minus 48 equals negative 4 open parentheses 2 x squared minus 25 x plus 12 close parentheses equals negative 4 open parentheses 2 x minus 1 close parentheses open parentheses x minus 12 close parentheses

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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.