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Factorising Quadratics (AQA GCSE Maths)
Revision Note
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
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 .
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)
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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
- 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
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 .
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)
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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
- 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
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
Recognise that and are both squared terms and the second term is subtracted from the first term - you can factorise using the difference of two squares.
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.
Recognise that and are both squared terms and the second term is subtracted from the first term - you can factorise using the difference of two squares.
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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