Key Stage 4 maths · Algebra

GCSE algebra: factorising quadratics, indices and identities

In Years 10 and 11 algebra moves from simple rearranging to manipulating expressions with confidence. Pupils factorise quadratic expressions, including the difference of two squares, simplify products and powers using the laws of indices, tell the difference between an equation and an identity, treat expressions as functions with inputs and outputs, and translate real situations into formulae and equations they can solve and interpret.

Factorising, indices and identities, explained

Factorising a quadratic is the reverse of expanding two brackets. To factorise x² + bx + c, a pupil looks for two numbers that multiply to give c and add to give b, then writes (x + p)(x + q). For x² + 7x + 12 the numbers are 3 and 4, so it becomes (x + 3)(x + 4). A special case is the difference of two squares: x² − 49 = (x + 7)(x − 7), and in general x² − a² = (x + a)(x − a). Pupils also meet these with surds, such as expanding (1 + √2)² = 3 + 2√2, and always check a factorisation by expanding it back out.

Simplifying sums, products and powers relies on the laws of indices. When multiplying powers of the same letter you add the indices, so x² × x³ = x⁵, and when dividing you subtract, so x⁶ ÷ x² = x⁴. A power of a power multiplies the indices: (x³)² = x⁶. Number parts are handled separately, so 3x² × 4x³ = 12x⁵ and (2x³)² = 4x⁶. Any non-zero number or letter to the power 0 is 1, and x⁻¹ means 1 ÷ x. These same laws are used with numbers, so 2⁵ × 2⁻² = 2³.

The difference between an equation and an identity matters in proofs. An equation such as 2x + 1 = 7 is true only for particular values, here x = 3. An identity such as 2(x + 3) ≡ 2x + 6 is true for every value of x, and is written with the symbol ≡. Pupils are expected to argue mathematically that two expressions are equivalent, normally by expanding and simplifying one or both sides, and to use algebra to support arguments, for instance showing that (n + 1)² − n² always equals 2n + 1, an odd number.

Functions are expressions seen as machines. For f(x) = 3x − 2, an input of 4 gives an output of f(4) = 3 × 4 − 2 = 10. Pupils also run the machine backwards: if the output is 13, solving 3x − 2 = 13 gives the input x = 5. For all pupils the aim is to understand the input and output idea. Inverse and composite functions, which reverse or combine function machines, are part of the higher content.

Finally, pupils translate situations into algebra. If a rectangle’s length is 3 cm more than its width w and its perimeter is 38 cm, then 2(w + w + 3) = 38. Expanding gives 4w + 6 = 38, so w = 8 and the length is 11 cm. Some problems give two unknowns and two facts, which pupils turn into simultaneous equations. In every case the solution has to be interpreted: a negative width would be rejected, and units must be stated.

Words your child will hear

Quadratic expression: An expression whose highest power is 2, such as x² + 5x − 14.
Factorise: Write an expression as a product, for example x² + 7x + 12 = (x + 3)(x + 4).
Difference of two squares: An expression of the form a² − b², which factorises as (a + b)(a − b).
Index laws: The rules for multiplying, dividing and raising powers, such as x² × x³ = x⁵.
Identity: A statement true for all values of the letters, written with ≡, such as 2(x + 3) ≡ 2x + 6.
Function: A rule that turns each input into exactly one output, such as f(x) = 3x − 2.
Formula: A rule linking quantities using letters, such as cost = 15 + 0.05 × minutes.

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Simplify and manipulate algebraic expressions (including those involving surds) by factorising quadratic expressions of the form x² + bx + c, including the difference of 2 squares
Simplify and manipulate algebraic expressions (including those involving surds) by simplifying expressions involving sums, products and powers, including the laws of indices
Know the difference between an equation and an identity; argue mathematically to show algebraic expressions are equivalent, and use algebra to support and construct arguments
Where appropriate, interpret simple expressions as functions with inputs and outputs
Translate simple situations or procedures into algebraic expressions or formulae; derive an equation (or 2 simultaneous equations), solve the equation(s) and interpret the solution

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

Factorise x² + 5x − 14, and check your answer.

A quadratic with a single x² term factorises into two brackets, (x + p)(x + q). We need two numbers that multiply to −14 and add to +5.
List the factor pairs of 14: 1 and 14, 2 and 7. The product is negative, so one number is positive and the other negative. Their sum must be +5, which rules out 1 and 14. For 2 and 7, the pair 7 and −2 has sum +5.
```
7 × (−2) = −14
7 + (−2) = 5
```
Write the brackets.
```
x² + 5x − 14 = (x + 7)(x − 2)
```
Check by expanding: multiply each term in the first bracket by each term in the second, then collect like terms.
```
(x + 7)(x − 2)
= x² − 2x + 7x − 14
= x² + 5x − 14
```

x² + 5x − 14 = (x + 7)(x − 2)

Common mistakes and how to help

Choosing numbers with the right product but the wrong sum, such as (x − 7)(x + 2)

Pupils find a pair that multiplies to −14 and stop, without checking the sum. Here the product is right but the middle term comes out as −5x.

What helps: Always expand the brackets to check. A helpful pattern: when the last number is negative, the larger of the two numbers carries the sign of the middle term.

Writing (x + 3)² = x² + 9

Squaring is applied to each term separately, as if the power could be shared out across the brackets.

What helps: Write the square out as (x + 3)(x + 3) and expand fully to get x² + 6x + 9. A square drawn with sides x + 3 splits into an x by x square, two 3 by x rectangles and a 3 by 3 square.

Multiplying the indices when multiplying terms, so 2x³ × 3x² = 6x⁶

The rule for a power of a power (multiply) gets confused with the rule for multiplying powers (add). Another slip is (3x)² = 3x² instead of 9x².

What helps: Write the terms out: x³ × x² = (x·x·x) × (x·x) = x⁵. For (3x)², write 3x × 3x = 9x², so everything inside the bracket is squared.

Thinking an identity is proved by trying one value

x² and 2x are both 4 when x = 2, which could suggest they are the same, but they differ at x = 3. One matching value only shows an equation holds there.

What helps: To show two expressions are identical, expand and simplify until both sides look the same. To show they are not, one value where they differ is enough.

How to help at home

Think-of-a-number tricks. Ask your child to think of a number, add 3, double it, subtract 6, then halve. They will always return to the number they chose. Ask them to prove it with algebra: (n + 3) × 2 − 6, divided by 2, simplifies to n.
Check by expanding, every time. Whenever your child factorises, ask them to multiply the brackets back out. It takes 20 seconds and turns a guess into a result they can trust.
Write a formula for something real. A taxi charges £3 plus £2 per mile; a phone contract is £15 plus 5p a minute over the allowance. Ask your child to write a formula, find the cost for a given amount, and work out what amount gives a given cost.
Draw the squares. Sketch a square with sides x + 3 on plain paper and split it into four pieces. Seeing x² + 6x + 9 as an area helps expanding and factorising make sense.

Builds on

Key Stage 3: Algebraic notation and formulae
Key Stage 3: Solving linear equations
Key Stage 3: Graphs of linear and quadratic functions
Key Stage 3: Sequences and the nth term