Factorise with sum and product

Find two numbers that multiply to the last term and add to the middle one.

The question

Factorise x² + 7x + 12

The trick

You need two numbers whose product is the last term and whose sum is the middle coefficient. Those two numbers go straight into the brackets.

Worked example

1Product needed: 12. Sum needed: 7
2Pairs that multiply to 12: 1×12, 2×6, 3×4
33 + 4 = 7 ✓
4So x² + 7x + 12 = (x + 3)(x + 4)

So: (x + 3)(x + 4)

Remember it like this

Multiply to the last, add to the middle. Always list the product pairs first — the sum then picks itself out.

In the exam

Watch the signs. A negative last term means one bracket is plus and one is minus. A negative middle with a positive last term means both brackets are minus.

Why it works

Expanding (x+a)(x+b) gives x² + (a+b)x + ab. The middle coefficient is always the sum and the last term is always the product — so factorising is just reading that backwards.

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Try it yourself

0/4 correct

1
Factorise x² + 5x + 6
Factorise x² + 5x + 6
2
Factorise x² + 8x + 15
Factorise x² + 8x + 15
3
Factorise x² − 7x + 12
Factorise x² − 7x + 12
4
Factorise x² + 2x − 8
Factorise x² + 2x − 8