Solving Quadratics by Factorising

Grade 5

The zero-product rule

If two things multiply together to give zero, then at least one of them must be zero. This is the zero-product rule, and it is the idea that makes solving by factorising work.

Once you write a quadratic as (x + p)(x + q) = 0, you know either (x + p) = 0 or (x + q) = 0. Each gives you one solution.

Worked example

EXAMPLE

Solve x² − 5x + 6 = 0.

Factorise: need two numbers that add to −5 and multiply to 6: that is −2 and −3. So (x − 2)(x − 3) = 0. Set each bracket to zero: x − 2 = 0 gives x = 2; x − 3 = 0 gives x = 3. x = 2 or x = 3.

First step: rearrange to zero

The equation must equal zero before you factorise. If you have x² + 3x = 10, rearrange to x² + 3x − 10 = 0 first. Never split a non-zero right-hand side across two brackets — that is not a valid move.

When factorising will not work

Some quadratics do not factorise neatly. If you cannot find a factor pair quickly, switch to the quadratic formula or completing the square. There is no shame in that — those methods always work.

Try this one

Solve by factorising.

\[ x^2 + 7x + 12 = 0 \]

[3]

Nova's hint:

This uses factorising a quadratic into two brackets.

Ask yourself: 'which two numbers multiply to give the constant term AND add to give the coefficient of \( x \)?'

For example, with \( x^2 + 5x + 6 = 0 \), you'd look for two numbers that multiply to 6 and add to 5, giving \( (x+2)(x+3)=0 \).

Try to Solve it with Nova?