Quadratic Equations

GCSE Maths · Algebra

Quadratic equations

A quadratic equation has the form ax² + bx + c = 0, where the highest power is x². It usually has two solutions (called roots), because a parabola can cross the x-axis twice. There are three main methods to solve them.

Method 1: Factorising

If the quadratic factorises, this is the quickest way.

1. Rearrange so one side is = 0.

2. Factorise into two brackets.

3. Set each bracket = 0 and solve.

Example: solve x² + 7x + 12 = 0.

Factorise: (x + 3)(x + 4) = 0.
So x + 3 = 0 → x = −3, or x + 4 = 0 → x = −4.
Solutions: x = −3 and x = −4.

(Why? If two things multiply to give 0, at least one must be 0.)

Method 2: The quadratic formula

Works for any quadratic (especially when it won't factorise):

x = [ −b ± √(b² − 4ac) ] ÷ 2a

Substitute a, b and c carefully (watch signs), and remember the ± gives the two roots.

Example: solve x² + 4x + 1 = 0 (a=1, b=4, c=1).

x = [ −4 ± √(16 − 4) ] ÷ 2 = [ −4 ± √12 ] ÷ 2

x ≈ −0.27 or x ≈ −3.73.

Method 3: Completing the square

Rewrite x² + bx + c as (x + b/2)² − (b/2)² + c. Useful for solving and for finding the turning point of the graph. E.g. x² + 6x + 5 = (x + 3)² − 4.

The discriminant

The part b² − 4ac tells you how many real solutions there are:

> 0 → two real solutions.
= 0 → one (repeated) solution.
< 0 → no real solutions.

Worked example

Solve x² − 5x + 6 = 0 by factorising.

Two numbers multiplying to 6, adding to −5 → −2 and −3.
(x − 2)(x − 3) = 0 → x = 2 or x = 3. ✓

Common mistakes

Forgetting there are usually two solutions.
Sign errors when substituting into the formula (especially with negative b or c).
Not rearranging to = 0 before factorising.

Exam tips

Try factorising first; use the formula if it doesn't factorise nicely.
Write the formula out and substitute clearly for method marks.
If asked to leave an answer in surd form, don't round.

Key facts to remember

Quadratic: ax² + bx + c = 0, usually two roots.
Solve by factorising (set each bracket = 0), the quadratic formula x = [−b ± √(b²−4ac)]/2a, or completing the square.
The discriminant b² − 4ac gives the number of real solutions (>0 two, =0 one, <0 none).

Quadratic Equations