Rukasha Maths

GCSE maths Beta

Quadratics: formula and completing the square · Higher notes

Higher · Algebra · AQA references A11, A18

Factorising is efficient when a quadratic has simple factors, but not every quadratic does. The quadratic formula works after the equation is written as ax² + bx + c = 0. Completing the square rewrites a quadratic in a form that reveals its turning point and also helps solve equations. Both methods preserve equality. Keep both signs when taking a square root: one positive square can come from a positive or negative number. Check each solution in the original equation.

Learning objectives

  • Solve quadratics with the quadratic formula.
  • Complete the square and identify a turning point.

Key ideas

Write the equation in standard form; keep coefficient signs.

ax2+bx+c=0,a≠0ax^2+bx+c=0,\qquad a\ne0

The formula gives both possible real roots.

x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Positive, zero or negative discriminants give two, one repeated or no real roots.

D=b2−4acD=b^2-4ac

Halve the linear coefficient, then correct the constant.

x2+bx=(x+b2)2−(b2)2x^2+bx=\left(x+\frac b2\right)^2-\left(\frac b2\right)^2

The completed-square form shows the turning point.

y=a(x−p)2+q⇒(p,q)y=a(x-p)^2+q\quad\Rightarrow\quad(p,q)

Worked examples

Example 1

Solve using the quadratic formula.

2x2+3x−2=0⇒a=2, b=3, c=−22x^2+3x-2=0\quad\Rightarrow\quad a=2,\ b=3,\ c=-2
D=32−4(2)(−2)=25D=3^2-4(2)(-2)=25
x=−3±254=−3±54x=\frac{-3\pm\sqrt{25}}4=\frac{-3\pm5}4

Answer

x=12orx=−2x=\frac12\quad\text{or}\quad x=-2

Example 2

Complete the square.

x2−8x+7=(x−4)2−16+7x^2-8x+7=(x-4)^2-16+7

Answer

(x−4)2−9(x-4)^2-9

Question styles to prepare for

Original practice guidance aligned to the specification, not a count of past-paper appearances or a prediction. The AO labels below describe the intended practice focus, not official AQA mark allocations.

AO1 · Methods

Solve a quadratic or complete the square to identify a turning point.

AO2 · Reasoning

Explain the number of solutions or justify why a root is invalid in context.

AO3 · Problem solving

Form a quadratic from an area or other relationship and interpret its roots.

When a task asks you to explain, include a mathematical reason. When solving a problem, show how the information becomes a calculation, equation or diagram, then interpret your result.

Avoid this mistake

Divide the whole numerator by the denominator. Keep both signs when taking a square root to solve an equation.

Try it yourself

Attempt each question before opening its answer.

Check 1: Complete the square for x² - 8x + 7.

Answer: (x - 4)² - 9

(x - 4)² = x² - 8x + 16, so subtract 9 to make the constant 7.

Check 2: Solve x² - 2x - 3 = 0.

Answer: x = 3 or x = -1

Factorise as (x - 3)(x + 1) = 0.

Explain it in your own words

  • Write the quadratic formula and label a, b and c.
  • Complete the square for x² + 6x + 5 and identify its turning point.
  • Explain why a square-root step can produce two solutions.

Keep practising

Download the compact topic guide (PDF) · Practise this topic in Blitz · All higher notes

Specification source

AQA Algebra specification · Assessment objectives. Checked 16 October 2026. Topic references identify related content; these notes do not cover every specification statement.