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

A right-angled triangle has legs of length \( x \) cm and \( (x + 2) \) cm, and a hypotenuse of \( 10 \) cm.

(a) Show that \( x^2 + 2x - 48 = 0 \). [3]

(b) Find the length of the shorter leg. [2]

Nova's hint:

This combines Pythagoras' theorem with solving a quadratic.

Ask yourself: 'how do the squares of the two legs relate to the hypotenuse?' Expand \( (x+2)^2 \) carefully, then collect terms.

Try to Solve it with Nova?