A8AlgebraFoundation & Higher

Quadratic Equations

A quadratic equation has an $x^{2}$ term. They usually have two solutions. There are three methods: factorising (quickest when it works), the quadratic formula (always works), and completing the square (Higher).

50 min Video by Maths Genie AQA GCSE Maths

Factorising QuadraticsWatch the walkthrough, then read the notes below.

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What you'll learn

Solve quadratics by factorising
Use the quadratic formula
Know when to use each method
Understand why there are two solutions

Solving by factorising

Rearrange so the equation equals zero, factorise, then set each bracket to zero. If $(x + 2) (x - 3) = 0$ , then $x = - 2$ or $x = 3$ — because if two things multiply to zero, one of them must be zero.

1Factorise:

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

2Each bracket = 0:

x = - 2

x = - 3

Answer

x = - 2

x = - 3

The quadratic formula

For $a x^{2} + b x + c = 0$ , the formula always gives the solutions. Substitute carefully, especially with negative numbers.

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

The quadratic formula — given in the exam.

Tip — Try factorising first; reach for the formula when factorising is hard or impossible.

Remember these

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

The quadratic formula.

(x + a) (x + b) = 0 \Rightarrow x = - a or x = - b

Zero-product rule.

Watch out for these

Solving before making the equation equal to zero.

Rearrange to “= 0” first, then factorise.

Giving only one solution.

A quadratic usually has two solutions — find both.

Key takeaways

Rearrange to = 0, then factorise and set each bracket to 0.
Quadratic formula always works: x = (−b ± √(b²−4ac))/2a.
Quadratics usually have two solutions.

Test yourself

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