Quadratic Equations

Algebra and Sequences

Why a zero product helps

If two quantities multiply to give zero then at least one of them must be zero, and this is the whole idea behind the method
Factorise the left-hand side into two brackets, then set each bracket equal to zero in turn
Each of those small equations is linear and gives one solution
Where the brackets are (x + m) and (x + n), the solutions are simply −m and −n, but that shortcut fails as soon as a bracket has a number in front of x

Exam tip

A quadratic has two solutions and both are wanted — 38 of the 43 mark schemes use "B1 for each", so one root still earns half. After factorising, set each bracket to zero separately rather than trying to read the answers straight off the brackets, which is where the signs get reversed.

Worked example

Two brackets with numbers in front of x

Solve 4x² − 4x − 15 = 0.

Solution:

The equation already has zero on one side, so factorise straight away
Two numbers multiplying to 4 × (−15) = −60 and adding to −4 are −10 and 6
Split the middle term: 4x² − 10x + 6x − 15 = 0
Group in pairs: 2x(2x − 5) + 3(2x − 5) = 0
Factorise: (2x − 5)(2x + 3) = 0
Set each bracket to zero: 2x − 5 = 0 gives x = 2.5, and 2x + 3 = 0 gives x = −1.5
The solutions are x = 2.5 or x = −1.5

When x itself is a factor

An equation with no constant term, such as 3x² = 7x, factorises to x(3x − 7) = 0
Treat the lone x as a bracket of its own, so it supplies the solution x = 0
Never divide both sides by x to tidy the equation up, because dividing by something that might be zero throws that solution away
Losing x = 0 in this way turns a two-mark answer into a one-mark answer