Completing the Square Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Completing the Square.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A technique for rewriting $ax^2 + bx + c$ in vertex form $a(x - h)^2 + k$ by adding and subtracting the value $\left(\frac{b}{2a}\right)^2$ to create a perfect square trinomial.

Imagine you have $x^2 + 6x$ and want a perfect square. A perfect square like $(x + 3)^2 = x^2 + 6x + 9$ needs that extra $+9$ . So you add 9 and subtract 9 to keep the expression equal—then group the perfect square part.

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How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Completing the square adds and subtracts $(\tfrac{b}{2})^2$ to turn $x^2+bx$ into a perfect square plus a leftover.

Common stuck point: The procedure for completing the square is the easy part; the trap is adding $(\tfrac{b}{2})^2$ without also subtracting it. Asking "Am I manufacturing a perfect-square trinomial by adding and subtracting the same value to keep it equal?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I manufacturing a perfect-square trinomial by adding and subtracting the same value to keep it equal?

Worked Examples

Example 1

medium

Rewrite

x^2 + 6x + 2

in vertex form by completing the square.

Answer

(x + 3)^2 - 7

First step

Take half of the coefficient of

x

\frac{6}{2} = 3

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Example 2

hard

Solve

x^2 - 4x - 5 = 0

by completing the square.

Example 3

medium

Rewrite

x^2 - 10x + 18

in vertex form by completing the square.

Example 4

medium

Rewrite

2x^2 - 12x + 5

in vertex form.

Example 5

hard

Solve

2x^2 + 8x - 10 = 0

by completing the square.

Example 6

hard

Solve

x^2 - 5x + 2 = 0

by completing the square; give exact answers.

Example 7

challenge

Use completing the square to derive the quadratic formula starting from

ax^2 + bx + c = 0

(

a \neq 0

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

What number completes the square for

x^2 + 10x + \,\_\,

Example 2

medium

Rewrite

x^2 - 8x + 20

in vertex form.

Example 3

easy

What value completes the square for

x^2 + 6x

Example 4

easy

What value completes the square for

x^2 + 10x

Example 5

easy

What value completes the square for

x^2 - 8x

Example 6

easy

Write

x^2 + 4x + 4

as a squared binomial.

Example 7

easy

Write

x^2 - 6x + 9

as a squared binomial.

Example 8

easy

Complete the square: rewrite

x^2 + 2x

in the form

(x+h)^2 + k

Example 9

easy

What is

h

when you complete the square on

x^2 - 10x

, writing

(x+h)^2 + k

Example 10

easy

x^2 + 5x + 6

a perfect-square trinomial?

Example 11

medium

Complete the square:

x^2 + 8x + 10

Example 12

medium

Complete the square:

x^2 - 6x + 1

Example 13

medium

Complete the square:

2x^2 + 8x + 5

Example 14

medium

Complete the square:

3x^2 - 12x + 7

Example 15

medium

Solve

x^2 + 6x + 5 = 0

by completing the square.

Example 16

medium

Solve

x^2 - 4x - 12 = 0

by completing the square.

Example 17

medium

Complete the square for

x^2 + 3x

(fractions allowed).

Example 18

medium

After completing the square,

x^2 - 2x + 7 = (x-1)^2 + c

. Find

c

Example 19

medium

Complete the square:

x^2 + 5x + 1

(fractions allowed).

Example 20

challenge

Find the minimum value of

f(x) = x^2 - 8x + 20

Example 21

challenge

For what value of

c

x^2 + 10x + c

a perfect-square trinomial?

Example 22

challenge

Derive the vertex of

y = ax^2 + bx + c

by completing the square; give the

x

-coordinate.

Example 23

easy

What number completes the square for

x^2 + 12x + \,\_\,

Example 24

easy

Write

x^2 - 14x + 49

as a squared binomial.

Example 25

easy

Complete the square: rewrite

x^2 + 4x

in the form

(x+h)^2 + k

Example 26

easy

x^2 + 6x + 9

a perfect-square trinomial?

Example 27

easy

Write

x^2 - 20x + 100

as a squared binomial.

Example 28

medium

Complete the square:

x^2 + 12x + 20

Example 29

medium

Complete the square:

4x^2 + 16x + 7

Example 30

medium

Solve

x^2 + 8x + 7 = 0

by completing the square.

Example 31

medium

Solve

x^2 - 10x + 21 = 0

by completing the square.

Example 32

medium

Complete the square for

x^2 + 7x

(fractions allowed).

Example 33

medium

Find the vertex of

y = x^2 - 6x + 11

Example 34

medium

Find the minimum value of

f(x) = x^2 + 4x + 9

Example 35

medium

Solve

x^2 + 4x - 1 = 0

by completing the square.

Example 36

hard

Find the vertex of

y = -2x^2 + 8x + 1

Example 37

hard

Rewrite

3x^2 + 6x - 1

in vertex form.

Example 38

hard

The path of a ball is

h(t) = -16t^2 + 64t + 5

. Find its maximum height by completing the square.

Example 39

challenge

Find all real values of

k

for which

x^2 + kx + 9

has exactly one real root.

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Related Concepts

Quadratic Standard Form Expressions Quadratic Vertex Form Quadratic Formula Conic Sections Overview

Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic standard formexpressions