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Completing the Square

A free Algebra II lesson from the “Transformations of Functions” unit, with a worked example and practice problems including step-by-step solutions.

Completing the square turns x^2 + bx into a perfect-square trinomial. Add (b/2)^2 to both sides or balance the expression so the equation stays equivalent.

What you'll learn

Why it matters: Converting standard form to vertex form, deriving the quadratic formula, and integrating in calculus all use completing the square. Adding the right constant turns a trinomial into a perfect-square binomial — geometrically, you are filling in the missing corner of a square.

Worked example

Problem. Complete the square for x^2 + 10x.

  1. Take half of 10, which is 5.
  2. Square 5.
  3. Add 25 to make x^2 + 10x + 25 = (x + 5)^2.

Answer: 25

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. What number completes the square for x^2 + 8x?

Show solution
  1. Half of 8 is 4.
  2. 4^2 = 16.

Answer: 16

2. What number completes the square for x^2 - 6x?

Show solution
  1. Half of -6 is -3.
  2. (-3)^2 = 9.

Answer: 9

3. x^2 + 12x + 36 factors as...

Choices: (x + 6)^2 · (x - 6)^2 · (x + 18)^2 · (x + 3)^2

Show solution
  1. Half of 12 is 6.

Answer: (x + 6)^2

4. x^2 - 14x + 49 factors as...

Choices: (x - 7)^2 · (x + 7)^2 · (x - 49)^2 · (x + 14)^2

Show solution
  1. Half of -14 is -7.

Answer: (x - 7)^2

5. Solve (x - 4)^2 = 25. Enter the positive solution.

Show solution
  1. x - 4 = 5 or -5.
  2. The positive solution is 9.

Answer: 9

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