Solving Quadratic Equations - IGCSE Maths (Factorising, the Formula & the Discriminant)

Quadratic equations appear on almost every IGCSE Maths paper, so solving them quickly and accurately is worth a lot of marks. This lesson covers the two methods you must know — factorising and the quadratic formula — plus the discriminant and what the solutions mean on a graph.

What is a quadratic equation?

A quadratic is any equation where the highest power of x is 2. In standard form it looks like ax² + bx + c = 0, where a, b and c are numbers and a ≠ 0. Your first step in any question is to rearrange it into this form.

What "solving" means

The solutions (or roots) of a quadratic are the values of x that make it equal to zero. On a graph, they are exactly where the U-shaped parabola crosses the x-axis. Most quadratics have two roots.

Method 1 — Factorising

Factorising uses one key idea: if two things multiply to give zero, at least one of them must be zero. When the coefficient of x² is 1, find two numbers that multiply to c and add to b.

Example: x² + 5x + 6 = 0. The numbers 2 and 3 multiply to 6 and add to 5, so it factorises to (x + 2)(x + 3) = 0, giving x = −2 or x = −3. Watch the signs: if the constant c is negative, your two numbers have opposite signs — e.g. x² − x − 6 = 0 factorises to (x − 3)(x + 2) = 0, so x = 3 or x = −2.

Method 2 — The quadratic formula

Some quadratics don't factorise neatly (try x² + 4x + 1 = 0). For those — and for any quadratic — use the formula:

x = ( −b ± √(b² − 4ac) ) / 2a

Substitute a, b and c carefully, using brackets so negatives behave. Example: 2x² + 3x − 2 = 0 has a = 2, b = 3, c = −2, giving x = (−3 ± √25)/4 = (−3 ± 5)/4, so x = ½ or x = −2.

The discriminant

The expression under the square root, b² − 4ac, is the discriminant. It tells you how many real solutions there are: positive → two solutions, zero → one repeated root, negative → no real roots (the curve never touches the x-axis).

Which method to use

Factorise when it's quick; use the formula when it's awkward or the question asks for answers to a set number of decimal places; complete the square only when the question tells you to.

Exam tips

Always rearrange to = 0 first, give both roots unless told otherwise, check by substituting back, watch your signs, and show every line of working to secure method marks.

🎥 Watch the full lesson: youtube.com/watch?v=yqSkkynucOI
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