Expanding and Factorising Quadratics
x

Revision notes for Edexcel GCSE Maths Expanding and Factorising Quadratics. Open the guide for explanations and worked examples. Written against the Edexcel GCSE Maths (1MA1) specification, so the content matches what's examinable rather than general Maths background.

Expanding and Factorising Quadratics

What you'll learn

  • Multiply out two brackets to make an expression with an x2x^2x2 term.
  • Collect like terms and handle negative signs carefully.
  • Write a quadratic back into brackets by finding number pairs.
  • Spot special patterns such as repeated brackets and a difference of two squares.

1. The key vocabulary

Before the main methods, make sure these words feel familiar.

Definition

Useful vocabulary

  • A term is a part of an expression separated by plus or minus signs. In 3x2−5x+73x^2 - 5x + 73x2−5x+7, the terms are 3x23x^23x2, −5x-5x−5x and the constant term 7.

  • A coefficient is the number multiplying a variable, such as 3 in 3x3x3x.

  • Like terms have the same variable and the same power, such as 6x6x6x and −2x-2x−2x.

  • A quadratic expression has a squared variable as its highest power, such as x2+4x−12x^2 + 4x - 12x2+4x−12.

  • To expand means to remove brackets by multiplying. To factorise means to put an expression back into brackets. A factor is one part being multiplied.

2. Collecting like terms

When you expand quadratics, you usually get two middle terms that can be collected.

Example

Collecting like terms

The expression is split into terms, with the two like terms in x shown as the pair to combine.

Simplify x2+7x−3x−21x^2 + 7x - 3x - 21x2+7x−3x−21.

  1. Identify the like terms: 7x7x7x and −3x-3x−3x both contain xxx.

  2. Collect them: 7x−3x=4x7x - 3x = 4x7x−3x=4x.

  3. Keep the other terms unchanged: x2+4x−21x^2 + 4x - 21x2+4x−21.

3. Expanding two brackets

A bracket like x+5x + 5x+5 has two terms. When you multiply two brackets, every term in the first bracket must multiply every term in the second bracket.

Key Idea

Four multiplications

Each term in the first bracket multiplies each term in the second bracket, giving four products.

For two brackets, make four products, then collect like terms. This stops you missing the middle terms.

Example

Expand and simplify (x+5)(x−4)

An area model shows the four products made when expanding (x + 5)(x - 4).

  1. Multiply the first terms: x×x=x2x \times x = x^2x×x=x2.

  2. Multiply the outside terms: x×−4=−4xx \times -4 = -4xx×−4=−4x.

  3. Multiply the inside terms: 5×x=5x5 \times x = 5x5×x=5x.

  4. Multiply the last terms: 5×−4=−205 \times -4 = -205×−4=−20.

  5. Collect the like terms: x2−4x+5x−20=x2+x−20x^2 - 4x + 5x - 20 = x^2 + x - 20x2−4x+5x−20=x2+x−20.

Common Mistake

Only multiplying the ends

Do not just multiply xxx with xxx and 5 with -4. The two middle multiplications are what create the xxx term.

4. Expanding when there are coefficients

If a term is like 3x3x3x, multiply the number parts and the letter parts. For example, 3x×2x=6x23x \times 2x = 6x^23x×2x=6x2.

Example

Expand and simplify (4x−1)(2x+3)

The grid keeps track of coefficients and signs in all four multiplications.

  1. Multiply 4x4x4x by 2x2x2x: 4x×2x=8x24x \times 2x = 8x^24x×2x=8x2.

  2. Multiply 4x4x4x by 3: 4x×3=12x4x \times 3 = 12x4x×3=12x.

  3. Multiply -1 by 2x2x2x: −1×2x=−2x-1 \times 2x = -2x−1×2x=−2x.

  4. Multiply -1 by 3: −1×3=−3-1 \times 3 = -3−1×3=−3.

  5. Collect the middle terms: 8x2+12x−2x−3=8x2+10x−38x^2 + 12x - 2x - 3 = 8x^2 + 10x - 38x2+12x−2x−3=8x2+10x−3.

Tip

Sign check

A negative times a positive is negative. A negative times a negative is positive. Say the signs out loud if you need to.

5. Squared brackets

A squared bracket means the bracket is multiplied by itself. So (a−6)2(a - 6)^2(a−6)2 means (a−6)(a−6)(a - 6)(a - 6)(a−6)(a−6).

Example

Expand and simplify (b−6)2

Squaring a bracket means using the same bracket on both sides of the grid.

  1. Rewrite it as two brackets: (b−6)(b−6)(b - 6)(b - 6)(b−6)(b−6).

  2. Multiply all four pairs: b2−6b−6b+36b^2 - 6b - 6b + 36b2−6b−6b+36.

  3. Collect the like terms: −6b−6b=−12b-6b - 6b = -12b−6b−6b=−12b.

  4. The answer is b2−12b+36b^2 - 12b + 36b2−12b+36.

Common Mistake

Missing the middle term

(b−6)2(b - 6)^2(b−6)2 is not b2+36b^2 + 36b2+36. You must expand it as two brackets.

6. Factorising simple quadratics

Factorising reverses expanding. For Grade 5 questions, many quadratics begin with x2x^2x2, so the brackets often start as (x  )(x  )(x \ \ )(x \ \ )(x  )(x  ).

Tip

Product and sum

For x2+bx+cx^2 + bx + cx2+bx+c, find two numbers with product ccc and sum bbb. Product means multiply; sum means add.

Example

Factorise x2+9x+20

A product-and-sum diagram identifies 4 and 5 as the numbers needed in the brackets.

  1. Start with two brackets because x2=x×xx^2 = x \times xx2=x×x: (x  )(x  )(x \ \ )(x \ \ )(x  )(x  ).

  2. Find two numbers that multiply to 20 and add to 9.

  3. The pair is 4 and 5, because 4×5=204 \times 5 = 204×5=20 and 4+5=94 + 5 = 94+5=9.

  4. Put them into the brackets: (x+4)(x+5)(x + 4)(x + 5)(x+4)(x+5).

7. Choosing the signs when factorising

The constant term is the number at the end. Its sign gives you a big clue:

  • Positive constant and positive middle term: both signs are positive.
  • Positive constant and negative middle term: both signs are negative.
  • Negative constant: one sign is positive and one is negative.
Example

Factorise n2−4n−21

The negative constant means one number is positive and one is negative, and the pair must add to -4.

  1. Start with (n  )(n  )(n \ \ )(n \ \ )(n  )(n  ).

  2. Look for two numbers that multiply to -21 and add to -4.

  3. The numbers are -7 and 3, because −7×3=−21-7 \times 3 = -21−7×3=−21 and −7+3=−4-7 + 3 = -4−7+3=−4.

  4. Put them into the brackets: (n−7)(n+3)(n - 7)(n + 3)(n−7)(n+3).

8. Special patterns

Difference of two squares

A difference of two squares is a subtraction where both parts are square terms.

a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b)a2−b2=(a−b)(a+b)
Example

Factorise x2−64

The expression is a difference of two squares: x^2 and 8^2.

  1. Recognise both parts as squares: x2x^2x2 is a square and 64 is 828^282.

  2. Use one minus bracket and one plus bracket.

  3. The answer is (x−8)(x+8)(x - 8)(x + 8)(x−8)(x+8).

Common Mistake

It must be a difference

x2+64x^2 + 64x2+64 is not a difference of two squares because there is no subtraction sign between the square terms.

Perfect square quadratics

Sometimes a quadratic comes from squaring one bracket, such as (x+6)2(x + 6)^2(x+6)2 or (x−6)2(x - 6)^2(x−6)2.

Example

Factorise x2−16x+64

  1. Notice that 64 is 828^282.

  2. Check the middle term: −8x−8x=−16x-8x - 8x = -16x−8x−8x=−16x.

  3. So the two brackets are the same: (x−8)(x−8)(x - 8)(x - 8)(x−8)(x−8).

  4. Write this more neatly as (x−8)2(x - 8)^2(x−8)2.

Exam technique

In the exam

  1. When expanding, make four multiplications before collecting like terms.

  2. When factorising, write down the product target and the sum target.

  3. Check your answer by quickly expanding your brackets again.

  4. Watch negative signs carefully, especially when the constant term is negative.

Self review

Check yourself

  • Can you expand (2x−3)(x+5)(2x - 3)(x + 5)(2x−3)(x+5) without missing any terms?

  • Can you factorise x2−x−42x^2 - x - 42x2−x−42 by finding a product and a sum?

  • Can you explain why x2−81x^2 - 81x2−81 is a difference of two squares?

Recap questions

Test yourself with 5 quick questions on this guide. Answer them all correctly to complete it.

You've reached the end

Test yourself on this topic, or move on to the next guide.

Practice questionsTake a quick quiz on this topicFlashcardsSelf-test with active recall
Solving QuadraticsUp next

How was this guide?

Expanding and Factorising Quadratics Revision Guide

  1. GCSE
  2. /Maths
  3. /Expanding and Factorising Quadratics