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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

Practice problems

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: What number completes the square for x^2 + 8x?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Half of 8 is 4.
  4. 4^2 = 16.
  5. Check the result by substituting or estimating: the response should match 16 and make sense in the original problem.

Answer: 16

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

Show solution
  1. Warm-up: First identify exactly what the question is asking: What number completes the square for x^2 - 6x?
  2. Choose the operation or relationship that matches the wording, then carry it out one clear step at a time.
  3. Half of -6 is -3.
  4. (-3)^2 = 9.
  5. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

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. Core Practice: First identify exactly what the question is asking: x^2 + 12x + 36 factors as...
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Half of 12 is 6.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

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. Core Practice: First identify exactly what the question is asking: x^2 - 14x + 49 factors as...
  2. Use the structure of the expression to choose a factoring pattern, then check that the factors multiply back to the original expression.
  3. Half of -14 is -7.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: (x - 7)^2

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

Show solution
  1. Challenge: First identify exactly what the question is asking: Solve (x - 4)^2 = 25. Enter the positive solution.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. x - 4 = 5 or -5.
  4. The positive solution is 9.
  5. Check the result by substituting or estimating: the response should match 9 and make sense in the original problem.

Answer: 9

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