Method 1
Expand the right-hand side and collect the $b x$ terms

$\begin{array}{rcl} {(x + b)}^{2} & = & (x + b) (x + b) \\ = & x^{2} + b x + b x + b^{2} \\ = & x^{2} + 2 b x + b^{2} \end{array}$

Make $x^{2} - 12 x + a$ equal to $x^{2} + 2 b x + b^{2}$ by comparing each term
The $x^{2}$ terms are equal
The $x$ terms are equal if

$- 12 = 2 b$

Solve this equation to find $b$ (by dividing both sides by 2)

$\begin{array}{rcl} - \frac{12}{2} & = & b \\ - 6 & = & b \end{array}$

$b = - 6$ [1]

The number terms on the end are equal if

$a = b^{2}$

[1]

Substitute in $b = - 6$

$a = {(- 6)}^{2} a = 36$

$a = 36$ [1]

Method 2
Complete the square on $x^{2} - 12 x + a$ by replacing $x^{2} - 12 x$ with ${(x - 6)}^{2} - 36$

$x^{2} - 12 x + a = {(x - 6)}^{2} - 36 + a$

${(x - 6)}^{2} - 36 + a$ must equal ${(x + b)}^{2}$
The brackets are equal if

$b = - 6$

$b = - 6$ [1]

The numbers outside the brackets are equal if

$- 36 + a = 0$

[1]

Solve the equation to find $a$ (by adding $36$ to both sides)

$a = 36$

$a = 36$ [1]

Completing the Square (AQA GCSE Maths)

Exam Questions

1. Number

Number Toolkit

Prime Factors, HCF & LCM

Powers, Roots & Standard Form

Fractions

Percentages

Simple & Compound Interest, Growth & Decay

Fractions, Decimals & Percentages

Rounding, Estimation & Bounds

Surds

Using a Calculator

2. Algebra

Algebra Toolkit

Algebraic Roots & Indices

Expanding Brackets

Factorising

Completing the Square

Algebraic Fractions

Rearranging Formulae

Algebraic Proof

Linear Equations

Solving Quadratic Equations

Simultaneous Equations

Iteration

Forming & Solving Equations

Functions

Coordinate Geometry

Linear Graphs y = mx + c

Graphs of Functions

Equation of a Circle

Estimating Gradients & Areas under Graphs

Real-Life Graphs

Solving Inequalities

Graphing Inequalities

Transformations of Graphs

Sequences

3. Ratio, Proportion & Rates of Change

Ratio Toolkit

Ratio Problem Solving

Direct & Inverse Proportion

Standard & Compound Units

Exchange Rates & Best Buys

4. Geometry & Measures

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Angles in Polygons & Parallel Lines

Bearings, Scale Drawing, Constructions & Loci

Circle Theorems

Area & Perimeter

Circles, Arcs & Sectors

Volume & Surface Area

Congruence, Similarity & Geometrical Proof

Area & Volume of Similar Shapes

Right-Angled Triangles - Pythagoras & Trigonometry

Sine, Cosine Rule & Area of Triangles

3D Pythagoras & Trigonometry

Vectors

Transformations

5. Probability

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Simple Probability Diagrams

Tree Diagrams

Combined & Conditional Probability

6. Statistics

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Statistical Diagrams

Histograms

Cumulative Frequency & Box Plots

Scatter Graphs & Correlation