Completing the Square · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Completing the Square

Mathematics · Algebra · ages 15-16
Name ______________________   Date ____________

  1. Write x squared + 4x + 9 as (x + p) squared + q. What is q?

    • 13
    • 5
    • -5
    • 17

  2. Write x squared + 2x as a completed square plus a constant. What is the constant you add and subtract?

    • 1
    • 2
    • 4
    • 0

  3. Write x squared + 6x + 2 as (x + p) squared + q. What is q?

    • 7
    • -13
    • 13
    • -7

  4. Which expression is a perfect square of the form (x + p) squared?

    • x squared + 5x + 25
    • x squared + 10x + 25
    • x squared + 10x + 50
    • x squared + 4x + 25

  5. Write x squared + 12x + 5 as (x + p) squared + q.

    • (x + 6) squared - 31
    • (x + 12) squared - 139
    • (x + 6) squared + 41
    • (x + 3) squared - 31

  6. Completing the square on x squared - 6x + 2 gives (x - 3) squared - 7.

    Circle one:   True   False

  7. Write x squared + 2x + 4 as (x + p) squared + q.

    • (x + 2) squared + 0
    • (x + 1) squared + 5
    • (x + 4) squared + 1
    • (x + 1) squared + 3

  8. Write 2x squared + 4x + 7 as a(x + p) squared + q.

    • 2(x + 1) squared + 5
    • 2(x + 2) squared + 3
    • 2(x + 1) squared + 3
    • 2(x + 2) squared + 5

  9. Write 4x squared + 12x + 5 as a(x + p) squared + q. What is q?

    Answer: ______________

  10. Write 3x squared + 6x + 1 as a(x + p) squared + q.

    • 3(x + 1) squared - 2
    • 3(x + 2) squared - 5
    • 3(x + 1) squared - 8
    • 3(x + 2) squared - 2
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Answer key

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Completing the Square W1-mt_rHmCfzB4Fm-s1

  1. 5 · Add and subtract 4: x squared + 4x + 4 - 4 + 9 is (x + 2) squared + 5, so q is 5.
  2. 1 · Half of 2 is 1, and 1 squared is 1, so you add and subtract 1.
  3. -7 · Add and subtract 9: x squared + 6x + 9 - 9 + 2 is (x + 3) squared - 7, so q is -7.
  4. x squared + 10x + 25 · In (x + 5) squared the middle term is 2 times 5, which is 10, and the constant is 5 squared, which is 25. So x squared + 10x + 25 fits.
  5. (x + 6) squared - 31 · Add and subtract 36: x squared + 12x + 36 - 36 + 5 is (x + 6) squared - 31.
  6. True · Add and subtract 9: x squared - 6x + 9 - 9 + 2 is (x - 3) squared - 7. The sign inside the square matches the sign of the x term.
  7. (x + 1) squared + 3 · Add and subtract 1: x squared + 2x + 1 - 1 + 4 is (x + 1) squared + 3.
  8. 2(x + 1) squared + 5 · Take out 2: 2(x squared + 2x) + 7. Inside, add and subtract 1: 2((x + 1) squared - 1) + 7, which is 2(x + 1) squared - 2 + 7, that is 2(x + 1) squared + 5.
  9. -4 · Take out 4: 4(x squared + 3x) + 5. Inside, add and subtract 9/4: 4((x + 3/2) squared - 9/4) + 5, which is 4(x + 3/2) squared - 9 + 5, that is 4(x + 3/2) squared - 4.
  10. 3(x + 1) squared - 2 · Take out 3: 3(x squared + 2x) + 1. Inside, add and subtract 1: 3((x + 1) squared - 1) + 1, which is 3(x + 1) squared - 3 + 1, that is 3(x + 1) squared - 2.
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