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Completing the Square

Completing the square rewrites a quadratic as a perfect square plus a constant: $a (x + p)^{2} + q$ . This instantly reveals the turning point of the curve and gives another way to solve equations — including ones that will not factorise.

35 min Video by Zeeshan Zamurred Quadratics

Edexcel AS Maths: 2.2 Completing The SquareWatch the full walkthrough before the notes below.

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What you'll be able to do

Complete the square for a quadratic where a = 1
Complete the square when the coefficient of x² is not 1
Use completed-square form to find the turning point
Solve quadratic equations by completing the square

The basic process (a = 1)

For $x^{2} + b x + c$ , halve the coefficient of $x$ , square it, and balance. The result is a bracket squared plus a leftover constant.

x^{2} + b x + c = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2} + c

Halve

b

for the bracket, then subtract the square of that half.

1Half of

6

3

, so start with

(x + 3)^{2}

(x + 3)^{2} = x^{2} + 6 x + 9

, which is

8

too big, so subtract

8

… then add the original

+ 1

3Result:

(x + 3)^{2} - 9 + 1

Answer

(x + 3)^{2} - 8

Tip — The number inside the bracket is always half the coefficient of x — no exceptions.

When a ≠ 1

If the coefficient of $x^{2}$ is not $1$ , first factor it out of the $x^{2}$ and $x$ terms, complete the square inside, then multiply back through.

1Factor

2

from the first two terms:

2 (x^{2} + 6 x) + 5

2Inside:

x^{2} + 6 x = (x + 3)^{2} - 9

3So

2 [(x + 3)^{2} - 9] + 5 = 2 (x + 3)^{2} - 18 + 5

Answer

2 (x + 3)^{2} - 13

Reading off the turning point

In completed-square form $a (x + p)^{2} + q$ , the turning point of the parabola is at $(- p, q)$ . The sign flips for the $x$ -coordinate.

y = a (x + p)^{2} + q \Rightarrow vertex at (- p, q)

Minimum point if

a > 0

, maximum point if

a < 0

1Here

p = - 4

and

q = - 7

, so the

x

-coordinate is

- (- 4) = 4

Answerturning point

(4, - 7)

Solving by completing the square

Once in completed-square form, isolate the bracket, square-root both sides (remembering $\pm$ ), and solve. This gives exact surd answers.

1Complete the square:

(x + 2)^{2} - 4 - 3 = 0

, i.e.

(x + 2)^{2} = 7

2Square root both sides:

x + 2 = \pm 7

Answer

x = - 2 \pm 7

Tip — Do not forget the ± when you square-root — it gives both solutions.

Formula recap

x^{2} + b x + c = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2} + c

Completing the square (a = 1).

a (x + p)^{2} + q

Completed-square form.

vertex = (- p, q)

Turning point read straight off.

Common mistakes to avoid

Forgetting to subtract (b/2)² after forming the bracket.

The bracket adds (b/2)², so you must subtract it back to keep the value the same.

Reading the vertex x-coordinate with the wrong sign.

For (x + p)², the vertex is at x = −p — the sign flips.

Key takeaways

Halve the coefficient of x for the bracket, then subtract its square.
If a ≠ 1, factor it out of the x² and x terms first.
Completed-square form a(x + p)² + q gives the vertex (−p, q) immediately.
Solve by isolating the bracket and square-rooting with ±.

Test yourself

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