Factorising quadratics is the opposite of expanding double brackets and is an essential skill that can be used to solve quadratic equations and sketch their graphs. Although it may seem difficult at first, remembering a few tips will help you to factorise most quadratics with ease!
Factorising Quadratics
What Does Factorising Quadratics Mean?
In the lesson on expanding double brackets we looked at multiplying two algebraic expressions such as (x + 3)(x + 7) which gives the quadratic expression x² + 10x + 21.
Factorising quadratics means to write a quadratic as the product of two brackets.
For example factorising x² + 10x + 21 means to write it in the form (x + 3)(x + 7). In this sense, factorising quadratics is the opposite of expanding two brackets.
Factorising Quadratics Examples
- x² + 8x + 15 = (x + 5)(x + 3)
- x² + 11x + 28 = (x + 4)(x + 7)
- x² – 3x – 10 = (x + 2)(x – 5)
If you’re confused by this don’t worry as we have plenty of examples to help you understand!
What is a Quadratic?
In order to understand factorising quadratics we first need to know what a quadratic is.
A quadratic expression is an algebraic expression which has a variable with a highest power of 2.
- x² + 5x + 6 is a quadratic expression as the highest power of x is 2.
- x³ + 3x² + 7 is not a quadratic as the highest power is 3.
- 3x + 7 is not a quadratic as the highest power of x is 1 (since 3x + 7 is the same as 3x¹ + 7).
How to Factorise Quadratics
First let’s look at some examples of expanding double brackets to see if we can spot any patterns.
(x + 3)(x + 5) = x² + 8x + 15
- In the above example where did the 15 come from?
- Where did the 8 come from?
(x + 7)(x – 2) = x² + 5x – 14
- In the above example where did the -14 come from?
- Where did the 5 come from?
You might have noticed that in both examples the last term in the quadratic comes from multiplying the constants together. The middle term comes from adding them together.
Factorising Quadratics Simple Method
For a quadratic in the form x² + bx + c:
- Step 1: Find all the factor pairs of b (all the pairs of numbers that multiply together to make b).
- Step 2: Of these factor pairs, find the pair which add together to make c.
- Step 3: Write these two numbers in the form (x + d)(x + e).
In general, to factorise a quadratic in the form x² + bx + c, you need to find two numbers that multiply together to make c and add together to make b.
Example 1
Question: Factorise x² + 7x + 12.
Step 1: Find all of the factor pairs of 12.
The factor pairs of 12 are:
- 1 and 12
- 2 and 6
- 3 and 4
Step 2: Find the pair which add together to make 7.
3 and 4 add to make 7.
Step 3: Write these two numbers in the form (x + d)(x + e).
Answer: x² + 7x + 12 = (x + 3)(x + 4)
Example 2
Question: Factorise x² + 11x + 30.
Step 1: The factor pairs of 30 are:
- 1 and 30
- 2 and 15
- 3 and 10
- 5 and 6
Step 2: 5 and 6 add to make 30.
Step 3: (x + 5)(x + 6)
Answer: x² + 11x + 30 = (x + 5)(x + 6)
Example 3: With a Negative Last Term
If the last term of the quadratic is negative, then one of the brackets must have a positive sign and the other must have a negative sign.
Ie. if the quadratic is in the form x² + bx – c then the double brackets must be in the form (x + d)(x – e).
Question: Factorise x² + 3x – 40.
Step 1: The factor pairs of 40 are:
- 1 and 40
- 2 and 20
- 4 and 10
- 5 and 8
Note: Since the final term is -40, one of these numbers must be positive and the other must be negative. They should add to make 3.
Step 2: -5 and 8 add to make 3.
Step 3: (x – 5)(x + 8)
Answer: x² + 3x – 40 = (x – 5)(x + 8)
Example 4: With a Negative Last Term
Question: Factorise x² + 3x – 28.
Step 1: The factor pairs of 28 are:
- 1 and 28
- 2 and 14
- 4 and 7
Note: Since the final term is -28, one of these numbers must be positive and the other must be negative. They should add to make 3.
Step 2: -4 and 7 add to make 3.
Step 3: (x – 4)(x + 7)
Answer: x² + 3x – 28 = (x – 4)(x + 7)
Example 4: Two Negative Terms
Since the last term is negative we still need to find one positive and one negative number. Since the middle term is also negative, the two numbers need to add to make a negative number.
Question: Factorise x² – 6x – 27.
Step 1: The factor pairs of 27 are:
- 1 and 27
- 3 and 9
Note: Since the final term is -27, one of these numbers must be positive and the other must be negative. They should also add to make -6.
Step 2: -9 and 3 add to make -6.
Step 3: (x – 4)(x + 7)
Answer: x² + 3x – 28 = (x – 4)(x + 7)
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