Algebra II

Completing the Square

Lesson

Completing the square rewrites a quadratic as a perfect square plus a constant — i.e. into vertex form:

x^2 + bx + c \;\longrightarrow\; (x + h)^2 + k

For monic quadratics (leading coefficient 1):

Take half of $b$ , call it $h$ .
The squared term becomes $(x + h)^2$ , which expands to $x^2 + 2hx + h^2$ .
Add and subtract $h^2$ to balance.
The constant in vertex form is $k = c - h^2$ .

Shortcut to remember: $h = b/2$ and $k = c - h^2$ .

Worked example 1

x^2 + 6x + 11

Half of 6 is 3, so $h = 3$ . Then $h^2 = 9$ , and $k = 11 - 9 = 2$ .

x^2 + 6x + 11 = (x + 3)^2 + 2

Worked example 2

x^2 - 8x + 5

Half of $-8$ is $-4$ , so $h = -4$ and the squared term is $(x - 4)^2$ . $h^2 = 16$ , and $k = 5 - 16 = -11$ .

x^2 - 8x + 5 = (x - 4)^2 - 11

How to type your answer

Write the answer in vertex form (x+h)^2+k or (x-h)^2+k. No spaces. If $k$ is negative, write a minus. Examples: (x+3)^2+2, (x-4)^2-11, (x+5)^2-7.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

x^2 + 2x + 3

Problem 2

x^2 + 4x + 5

Problem 3

x^2 - 2x + 5

Problem 4

x^2 + 6x + 10

Practice

Standard problems matching the lesson.

Problem 5

x^2 + 6x + 11

Problem 6

x^2 - 8x + 5

Problem 7

x^2 + 4x - 1

Problem 8

x^2 - 6x + 13

Problem 9

x^2 + 10x + 21

Problem 10

x^2 - 10x + 30

Problem 11

x^2 + 8x

Problem 12

x^2 - 4x

Problem 13

x^2 + 12x + 32

Problem 14

x^2 - 14x + 50

Problem 15

x^2 + 4x + 7

Problem 16

x^2 + 16x + 60

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 17

x^2 - 12x + 27

Problem 18

x^2 + 14x + 30

Problem 19

x^2 - 16x + 70

Problem 20

x^2 + 20x + 90

Problem 21

x^2 - 18x + 80

Problem 22

x^2 + 6x - 16

Practice

Standard problems matching the lesson.

Problem 23

Trajectory y = x² − 6x + 13. Write in vertex form.

Problem 24

Profit model y = x² + 4x + 7. Write in vertex form.

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 25

Projectile path y = x² − 10x + 30. Write in vertex form.

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