Quadratic Equations

Quadratic equations have the form ax² + bx + c = 0, and their graph is a parabola. In IGCSE there are two core methods: factorising and the quadratic formula.

1. Factorising

Find two numbers whose product is a·c and whose sum is b.

x² + 5x + 6 = 0

Numbers with product 6 and sum 5: 2 and 3

(x + 2)(x + 3) = 0

x = −2 or x = −3

2. The quadratic formula

Always works, even when the expression does not factorise:

x = ( −b ± √(b² − 4ac) ) / 2a

Example: 2x² − 4x − 3 = 0 → a = 2, b = −4, c = −3

x = (4 ± √(16 + 24)) / 4 = (4 ± √40) / 4 → x ≈ 2.58 or x ≈ −0.58

The discriminant b² − 4ac tells you the number of roots: > 0 two roots, = 0 one root, < 0 no real roots.

Worked example

Solve x² − 7x + 12 = 0.

Numbers with product 12 and sum −7: −3 and −4 →

(x − 3)(x − 4) = 0 → x = 3 or x = 4

Exam tip

If the question says "give your answer to 2 decimal places", the quadratic formula is almost certainly expected — don't waste time trying to factorise. For 0606 (Additional Maths), the same formula underpins later work on differentiation and graphs, so learn the reasoning, not just the steps.