Factorising Quadratics (AQA GCSE Further Maths)
Revision Note
Author
Jamie WExpertise
Maths
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. , it can be called a “monic” quadratic expression
- If a ≠ 1 e.g. , it can be called a “non-monic” quadratic expression
Method 1: Factorising "by inspection"
- This is shown easiest through an example; factorising
- We need a pair of numbers that for
- 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:
- multiply to c
Method 2: Factorising "by grouping"
- This is shown easiest through an example; factorising
- We need a pair of numbers that for
- 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
- 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
- Note that these now have a common factor of (x + 2) so this whole bracket can be factorised out
- multiply to c
Method 3: Factorising "by using a grid"
- This is shown easiest through an example; factorising
- We need a pair of numbers that for
- 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
- multiply to c
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
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 a ≠ 1 so you should learn at least one of them too
Exam 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 .
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 .
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 .
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)
Factorising Harder Quadratics
How do I factorise a harder quadratic expression?
Factorising a ≠ 1 "by grouping"
- This is shown easiest through an example; factorising
- We need a pair of numbers that for
- 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
- 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
- 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
- multiply to ac
Factorising a ≠ 1 "by using a grid"
- This is shown easiest through an example; factorising
- We need a pair of numbers that for
- 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
- multiply to ac
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
- We can now read-off the factors from the column and row headings
How do I factorise a quadratic with two variables?
- To factorise 3x2 + 13xy - 10y2
- Factorise the easier quadratic 3x2 + 13x - 10
- (3x - 2)(x + 5)
- Insert y's on the last terms in the brackets
- (3x - 2y)(x + 5y)
- Factorise the easier quadratic 3x2 + 13x - 10
- Check by expanding (3x - 2y)(x + 5y)
- 3x2 + 15xy - 2yx - 10y2
- 3x2 + 13xy - 10y2 ✓
- 3x2 + 15xy - 2yx - 10y2
Exam 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 .
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 .
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)
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
- for example,
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
- 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
- but the a and the b cannot swap positions
- It might not be obvious that you can use the difference of two squares
- Try factoring out any common factors first
Exam 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
(i)
Factorise fully .
(ii)
Factorise .
(i)
The highest common factor of and is , so take this out as a factor
is a difference of two squares, as and
We can factorise the bracket into two further brackets using the difference of two squares
(ii)
is a difference of two squares, as both brackets are squared (and one is being subtracted from the other)
Use the pattern to help you
Here, and
Simplifying
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