AlgebraGCSE Higher

Factorising quadratics - in the form

Factorise a quadratic using the split method.

  • Compare the expression with : is the coefficient of , the coefficient of , and the constant.
  • The split method uses two numbers that multiply to and add to .
  • Split the middle term, factorise each pair, then take out the repeated bracket.

Example 1

Factorise fully.

Compare with .

Find two numbers that multiply to make and add to make .

The product and sum are both positive, so both numbers must be positive.

Selected factor-pair trials, their products and their sums. The chosen pair is circled.
NumbersProductSum
Chosen pair

Split as , then factorise each pair.

Split the middle term.Group the first pair and the second pair.Factorise each pair, then take out the repeated bracket.

Example 2

Factorise fully.

Compare with .

Find two numbers that multiply to make and add to make .

A negative product means one number is positive and one is negative.

Selected factor-pair trials, their products and their sums. The chosen pair is circled. Vertical dots indicate other factor pairs that are not shown.
NumbersProductSum
Chosen pair

Split as , then factorise each pair.

Split the middle term.Group the first pair and the second pair.Factorise each pair, then take out the repeated bracket.

Remember
Take out any common numerical factor before using this method.

Your turn

Question 1

Factorise fully.

Check answer

Compare with .

Find two numbers that multiply to make and add to make .

The product and sum are both positive, so both numbers must be positive.

Selected factor-pair trials, their products and their sums. The chosen pair is circled.
NumbersProductSum
Chosen pair

Split as , then factorise each pair.

Split the middle term.Group the first pair and the second pair.Factorise each pair, then take out the repeated bracket.
Question 2

Factorise fully.

Check answer

Compare with .

Find two numbers that multiply to make and add to make .

The product and sum are both positive, so both numbers must be positive.

Selected factor-pair trials, their products and their sums. The chosen pair is circled.
NumbersProductSum
Chosen pair

Split as , then factorise each pair.

Split the middle term.Group the first pair and the second pair.Factorise each pair, then take out the repeated bracket.
Question 3

Factorise fully.

Check answer

Compare with .

Find two numbers that multiply to make and add to make .

A negative product means one number is positive and one is negative.

Selected factor-pair trials, their products and their sums. The chosen pair is circled. Vertical dots indicate other factor pairs that are not shown.
NumbersProductSum
Chosen pair

Split as , then factorise each pair.

Split the middle term.Group the first pair and the second pair.Factorise each pair, then take out the repeated bracket.
Question 4

Factorise fully.

Check answer

Compare with .

Find two numbers that multiply to make and add to make .

A negative product means one number is positive and one is negative.

Selected factor-pair trials, their products and their sums. The chosen pair is circled. Vertical dots indicate other factor pairs that are not shown.
NumbersProductSum
Chosen pair

Split as , then factorise each pair.

Split the middle term.Group the first pair and the second pair.Factorise each pair, then take out the repeated bracket.
Question 5

Factorise fully.

Check answer

Compare with .

Find two numbers that multiply to make and add to make .

A positive product means matching signs. The negative sum tells us both numbers must be negative.

Selected factor-pair trials, their products and their sums. The chosen pair is circled. Vertical dots indicate other factor pairs that are not shown.
NumbersProductSum
Chosen pair

Split as , then factorise each pair.

Split the middle term.Group the first pair and the second pair.Factorise each pair, then take out the repeated bracket.
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