MathematicsChapter 2: Quadratic Equations

Completing the Square

For x^2 + px, add (p/2)^2 to make (x + p/2)^2.

Learning objectives

Explain completing the square using the correct definitions, conditions, and notation.
Apply add (half of b)^2, then square root correctly in routine and exam-style questions.
Show each important step, simplify the result, and check that it satisfies the original conditions.

Definition

For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadratics that do not factor nicely, and it is how the formula is born.

Mathematical form

x = 2 + sqrt(3) or x = 2 - sqrt(3)

x = 2 + sqrt(3) or x

Where / symbols

COMPLETING THE SQUARE: For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadratics that do not factor nicely, and it is how the formula is born.
THE NUMBER TO ADD: x^2 + 6x needs 3^2 = 9. x^2 - 4x needs (-2)^2 = 4. Always square half of the x-coefficient (when a=1).
SOLVE: x^2 - 4x + 1 = 0 -> (x-2)^2 - 4 + 1 = 0 -> (x-2)^2 = 3 -> x-2 = +/- sqrt(3) -> x = 2 +/- sqrt(3).

Overview

Half, square, move:

The Number To Add

x^2 + 6x needs 3^2 = 9. x^2 - 4x needs (-2)^2 = 4. Always square half of the x-coefficient (when a=1).

Examples x^2+6x -> +9 · x^2-4x -> +4 · Not +6 or +16

Solve

x² - 4x + 1 = 0 -> (x-2)² - 4 + 1 = 0 -> (x-2)² = 3 -> x-2 = +/- sqrt(3) -> x = 2 +/- sqrt(3).

Examples (x-2)^2 = 3 · x = 2 +/- sqrt(3) · Two real roots

WHAT — COMPLETING THE SQUARE

For x^2 + px, add (p/2)^2 to make (x + p/2)^2. Then move the leftover constant. This solves quadratics that do not factor nicely, and it is how the formula is born. The symbols and conditions are part of the mathematics, so include them when stating the rule.

WHY — Why this topic matters

Completing The Square matters because quadratic equations model areas, motion, and many algebraic relationships. Knowing more than one solution method lets you choose the quickest reliable method and check the nature of the roots.

HOW — How to solve and explain it

Start by identifying what is given and what must be found. Work through THE NUMBER TO ADD, then SOLVE. Worked: solve x^2 - 4x + 1 = 0 by completing the square. x^2 - 4x + 1 = 0 x^2 - 4x = -1 half of -4 is -2, and (-2)^2 = 4 x^2 - 4x + 4 = -1 + 4 (x - 2)^2 = 3 x - 2 = +/- sqrt(3) x = 2 + sqrt(3) or x = 2 - sqrt(3) Finish by checking signs, restrictions, units, and whether the answer matches the question.

Worked example

solve x^2 - 4x + 1 = 0 by completing the square.
x² - 4x + 1 = 0
x² - 4x = -1
half of -4 is -2, and (-2)² = 4
x² - 4x + 4 = -1 + 4
(x - 2)² = 3
x - 2 = +/- sqrt(3)
x = 2 + sqrt(3) or x = 2 - sqrt(3)

In simple words

If a is not 1, divide the whole equation by a first, then complete the square.

Key points

add (half of b)^2
then square root
(b/2)² if a=1
isolate the square

Exam preparation

Exam Focus

Quick revision

Revision Zone

Review the key facts, formulas and checks before you move on.