How to Solve Quadratic Equations for IGCSE (CIE) Maths

How to Solve Quadratic Equations for IGCSE (CIE) Maths

Factorise, use the formula, complete the square, or sketch it: four completely different routes to the same two numbers, and picking the slow one under exam pressure is how a three-mark question quietly eats six minutes.

None of the four methods is hard on its own. What separates a fast, clean answer from a slow, messy one is recognising which method the numbers, or the question's own wording, are actually asking for.

This guide walks through all four with the same worked examples an examiner would recognise from a mark scheme. It sits inside the Algebra chapter of the CIE IGCSE Extended syllabus, alongside sequences, inequalities, and simultaneous equations. If you haven't locked down solving two equations at once yet, our guide to simultaneous equations covers that separately.

ax² + bx + c = 0: Naming a, b and c Correctly First

The general form is ax² + bx + c = 0, where a, b, and c are constants and a cannot be zero. If a = 0 there's no x² term left, and it's just a linear equation instead. a is the coefficient of x², b is the coefficient of x, and c is the constant term.

Every method below starts by identifying these three values correctly. Get a, b, or c wrong and every method fails from that point on, no matter how clean the rest of the working is.

Four Methods, One Answer: Which One Actually Fits

CIE IGCSE Extended expects you to solve quadratics by factorisation, by the quadratic formula, by completing the square, and by graphing. They all find the same answer, the values of x where the equation equals zero, but they suit different situations, and exam questions often tell you which one to use through their own wording.

Method

Best used when

Pros

Cons

Factorisation

The equation factorises neatly (integer or simple fraction roots)

Fastest, no calculator needed, shows working clearly

Doesn't work if the equation doesn't factorise nicely

Quadratic formula

The question says "give your answer to 2 decimal places," or factorising isn't obvious

Always works, for any a, b, c

More arithmetic; easy to slip on negative signs under the root

Completing the square

You need the vertex of the graph, or the question specifies this method

Reveals the vertex (turning point) directly; proves the formula

Slower for simple equations; fiddly with a ≠ 1

Graphing

The question gives or asks for a sketch/table of values

Visual, shows the shape of the curve and number of roots

Reading exact values off a graph is imprecise unless roots are whole numbers

Factorisation: Finding Two Numbers That Multiply and Add

Factorisation means finding two numbers that multiply to give a × c and add to give b, then rewriting the equation as two linear brackets multiplied together. If two brackets multiply to zero, one of them must equal zero. That's the zero-product rule, and it's what the whole method rests on.

Worked example: Solve x² − x − 6 = 0.

Here a = 1, b = −1, c = −6, so a × c = −6. You need two numbers that multiply to −6 and add to −1: those are −3 and 2.

Rewrite: (x − 3)(x + 2) = 0.

By the zero-product rule, x − 3 = 0 or x + 2 = 0, so x = 3 or x = −2.

When a is not 1, split the middle term. Solve 2x² + 7x + 3 = 0.

a × c = 2 × 3 = 6. You need two numbers multiplying to 6 and adding to 7: those are 6 and 1.

Rewrite the middle term: 2x² + 6x + x + 3 = 0.

Group and factor: 2x(x + 3) + 1(x + 3) = 0 → (2x + 1)(x + 3) = 0.

So x = −½ or x = −3.

The Quadratic Formula, and the One Phrase That Tells You to Use It

The quadratic formula is x = [−b ± √(b² − 4ac)] / 2a, and it solves any quadratic equation, whether or not it factorises. Reach for it whenever the question specifically asks for an answer to two decimal places. That instruction is the exam's way of telling you the roots aren't nice whole numbers, so factorisation would just waste time.

Worked example: Solve x² − x − 6 = 0 using the formula.

a = 1, b = −1, c = −6.

x = [−(−1) ± √((−1)² − 4(1)(−6))] / 2(1) = [1 ± √(1 + 24)] / 2 = [1 ± √25] / 2 = [1 ± 5] / 2.

So x = 3 or x = −2, matching the factorisation answer exactly. That's a genuinely useful way to check your own working whenever a question happens to factorise as well.

Completing the Square: Solving and Finding the Vertex at Once

Completing the square rewrites ax² + bx + c in the form a(x + p)² + q, which lets you solve the equation and also read off the coordinates of the parabola's turning point (vertex) directly, since for a(x + h)² + k the vertex is at (−h, k).

Worked example: Solve x² + 6x + 5 = 0 by completing the square.

Take half of the coefficient of x (6 ÷ 2 = 3) and square it (3² = 9).

Add and subtract 9: x² + 6x + 9 − 9 + 5 = 0.

The first three terms are a perfect square: (x + 3)² − 4 = 0.

So (x + 3)² = 4 → x + 3 = ±2 → x = −1 or x = −5.

Here the vertex form is (x + 3)² − 4, so the vertex of y = x² + 6x + 5 is at (−3, −4). That's a free extra answer if the question also asks for the minimum point, at no extra working.

This is also exactly how the quadratic formula itself gets derived: complete the square on the general equation ax² + bx + c = 0 and you arrive at x = [−b ± √(b² − 4ac)] / 2a. If you ever blank on the formula in an exam, you can rebuild it this way from scratch.

Graphing: Reading the Roots Straight Off the Curve

Graphing means plotting y = ax² + bx + c and reading off the x-intercepts, the points where the curve crosses the x-axis, where y = 0, because those x-values are exactly the solutions to ax² + bx + c = 0.

For x² − x − 6 = 0, the graph of y = x² − x − 6 crosses the x-axis at x = −2 and x = 3, matching both algebraic methods above. Graphing is most useful when a question gives you a table of values to plot, or asks how many real roots an equation has: two crossings, one touching point, or none at all.

Picking the Right Method Under Exam Conditions

Read the question wording first. "Solve by factorisation" or "solve by completing the square" tells you the method directly, and marks are awarded for using it even if another method would get the right answer faster.

If the method isn't specified, try factorisation first since it's quickest. Switch to the quadratic formula if the numbers don't factorise or the question asks for two decimal places. Use completing the square when you need the vertex. Use graphing only when a sketch or table of values is explicitly requested.

Frequently Asked Questions

What is the general form of a quadratic equation? The general form is ax² + bx + c = 0, where a, b, and c are numbers and a ≠ 0. a is the coefficient of x², b is the coefficient of x, and c is the constant term. Every solving method starts by identifying these three values.

Which method should I use for a CIE IGCSE quadratic equation? Use whichever method the question names directly, since marks are awarded for the method itself. If no method is specified, try factorisation first for speed, switch to the quadratic formula when the numbers don't factorise neatly or two decimal places are requested, and use completing the square when the question asks for the vertex or turning point.

Why does a question ask for the answer "to 2 decimal places"? That phrase is a signal that the roots aren't whole numbers or simple fractions, so factorisation won't work cleanly. It's a direct instruction to use the quadratic formula, x = [−b ± √(b² − 4ac)] / 2a, and round your final answer, not the numbers inside the calculation itself.

Can every quadratic equation be factorised? No. Only quadratics with rational roots factorise neatly into two linear brackets. The quadratic formula and completing the square both work for every quadratic equation, including ones with irrational or no real roots, which is exactly why they're the reliable fallback methods.

What's the difference between completing the square and the quadratic formula? Completing the square rewrites the equation as a(x + p)² + q, which both solves it and reveals the vertex (−p, q) of the parabola. The quadratic formula, x = [−b ± √(b² − 4ac)] / 2a, is actually derived by completing the square on the general equation. It's faster for solving but doesn't hand you the vertex directly.

Before You Pick Up a Pen

Before you start solving, spend five seconds deciding: does the question name a method, do the numbers look like they'll factorise cleanly, or is "2 decimal places" sitting right there in the wording? That five-second check is what actually saves the time, not solving faster once you've already committed to the wrong route.

Work through a few past paper quadratics timing yourself on the decision, not just the algebra, and see how often your first instinct matches what the mark scheme rewards. If you want to check whether a full worked solution would score full marks for method as well as the final answer, the AI Tutor can review it against the same criteria examiners use.