Quadratic Equations

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CIE IGCSE International Maths — Extended

Quadratic Equations

💡 The Big Idea: Any equation with an x² in it can be solved by getting everything equal to zero, then either breaking it into two simple brackets or plugging it straight into one reliable formula — either way, you're just hunting for the x-values that make the whole thing equal zero.

Chapter Summary

A quadratic equation has the form ax² + bx + c = 0, where a ≠ 0 — the highest power of x is 2.
Factorising means rewriting the quadratic as two brackets multiplied together, then using the "zero product rule" — if two things multiply to give 0, one of them must BE 0.
Every quadratic can be solved with the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a — this works even when factorising is impossible.
The discriminant (b² − 4ac) — the bit under the square root — tells you in advance how many solutions you'll get: 2, 1, or 0.
Graphic display calculators can solve quadratics directly using a built-in polynomial solver — but you still need to know the method for non-calculator papers.
Watch out for the classic trap: when x is a common factor (like x(x − 4) = 0), never divide both sides by x — you'll silently lose the x = 0 solution.

1. Solving Quadratics by Factorising

Think of factorising as reverse-expanding. Normally you'd multiply out brackets like (x − 2)(x + 5) to get x² + 3x − 10. Factorising does the opposite: you're given x² + 3x − 10 = 0, and your job is to find the two brackets that would multiply together to make it.

Why bother? Because once it's in the form (something)(something) = 0, you can use a genuinely powerful trick:

The Zero Product Rule

If two numbers multiply together to give zero, one of them must be zero. There's no other way to get zero from a multiplication (other than 0 × 0). So if (x − 3)(x + 7) = 0, then either x − 3 = 0 or x + 7 = 0 — and you solve each bracket separately as its own mini equation.

A The Basic Method

Rearrange the equation into the form ax² + bx + c = 0 — get zero onto one side. It's easier if you arrange things so that a (the coefficient of x²) is positive.
Factorise the left-hand side into two brackets.
Set each bracket equal to zero separately, and solve each one as its own simple equation.
Write both solutions together using "or" — a quadratic almost always has two answers.

Worked Example

Solve (x − 3)(x + 7) = 0

x − 3 = 0 → x = 3 (add 3 to both sides)

x + 7 = 0 → x = −7 (subtract 7 from both sides)

Two solutions: x = 3 or x = −7

Spot the pattern

When the coefficient of x² is just 1 (like in the example above), the two solutions are simply the numbers in the brackets with the opposite sign. (x − 3) gave x = +3. (x + 7) gave x = −7. This is a handy sanity check, but it only works when a = 1.

B When There Are Numbers in Front of the x's

The process is exactly the same — but you can no longer just flip the sign to get the answer. There's real work to do in each bracket, because now the bracket doesn't just say "x + something", it says "(a number) × x + something".

Worked Example

Solve (2x − 3)(3x + 5) = 0

2x − 3 = 0 → 2x = 3 → x = 3/2 (add 3, then divide by 2)

3x + 5 = 0 → 3x = −5 → x = −5/3 (subtract 5, then divide by 3)

Two solutions: x = 3/2 or x = −5/3

C

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Also in the full note

2. The Quadratic Formula
3. Solving Quadratics Using a Calculator
What to Memorise
Concepts Checklist
Exam Tips & Common Mistakes
C When x Itself Is a Factor
A How to Use It
B Giving Answers in Exact (Surd) Form

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Quadratic Equations