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Math Grade 9-12 Answer Key

Math: Completing the Square

Rewrite quadratic expressions and solve quadratic equations

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Math: Completing the Square

Rewrite quadratic expressions and solve quadratic equations

Math - Grade 9-12

Instructions: Read each problem carefully. Complete the square step by step and simplify your final answer.
  1. 1

    Rewrite x^2 + 6x + 5 in the form (x + a)^2 + b.

    Take half of 6 and square it.

    The expression rewrites as (x + 3)^2 - 4 because x^2 + 6x + 5 = x^2 + 6x + 9 - 9 + 5.
  2. 2

    Rewrite x^2 - 10x + 7 in the form (x + a)^2 + b.

    The expression rewrites as (x - 5)^2 - 18 because x^2 - 10x + 7 = x^2 - 10x + 25 - 25 + 7.
  3. 3

    Rewrite x^2 + 4x - 1 in the form (x + a)^2 + b.

    Find the number that makes the trinomial a perfect square.

    The expression rewrites as (x + 2)^2 - 5 because x^2 + 4x - 1 = x^2 + 4x + 4 - 4 - 1.
  4. 4

    Rewrite x^2 - 8x + 3 in the form (x + a)^2 + b.

    The expression rewrites as (x - 4)^2 - 13 because x^2 - 8x + 3 = x^2 - 8x + 16 - 16 + 3.
  5. 5

    Solve by completing the square: x^2 + 6x + 1 = 0.

    Move the constant term first, then complete the square.

    The solutions are x = -3 + 2sqrt(2) and x = -3 - 2sqrt(2). Completing the square gives (x + 3)^2 = 8.
  6. 6

    Solve by completing the square: x^2 - 4x - 12 = 0.

    The solutions are x = 6 and x = -2. Completing the square gives (x - 2)^2 = 16.
  7. 7

    Solve by completing the square: x^2 + 2x - 7 = 0.

    After forming a square, take the square root of both sides.

    The solutions are x = -1 + 2sqrt(2) and x = -1 - 2sqrt(2). Completing the square gives (x + 1)^2 = 8.
  8. 8

    Solve by completing the square: x^2 - 12x + 20 = 0.

    The solutions are x = 10 and x = 2. Completing the square gives (x - 6)^2 = 16.
  9. 9

    Solve by completing the square: x^2 + 8x + 3 = 0.

    Half of 8 is the key value to use.

    The solutions are x = -4 + sqrt(13) and x = -4 - sqrt(13). Completing the square gives (x + 4)^2 = 13.
  10. 10

    Rewrite 2x^2 + 12x + 7 in vertex form a(x - h)^2 + k.

    The expression rewrites as 2(x + 3)^2 - 11 because 2x^2 + 12x + 7 = 2(x^2 + 6x) + 7 = 2(x^2 + 6x + 9) + 7 - 18.
  11. 11

    Rewrite 3x^2 - 18x + 10 in vertex form a(x - h)^2 + k.

    Factor the leading coefficient from the x^2 and x terms first.

    The expression rewrites as 3(x - 3)^2 - 17 because 3x^2 - 18x + 10 = 3(x^2 - 6x) + 10 = 3(x^2 - 6x + 9) + 10 - 27.
  12. 12

    Rewrite -x^2 + 4x + 1 in vertex form a(x - h)^2 + k.

    The expression rewrites as -(x - 2)^2 + 5 because -x^2 + 4x + 1 = -(x^2 - 4x) + 1 = -(x^2 - 4x + 4) + 1 + 4.
  13. 13

    Solve by completing the square: 2x^2 + 8x - 6 = 0.

    Start by dividing both sides by 2.

    The solutions are x = 1 and x = -5. Dividing by 2 gives x^2 + 4x - 3 = 0, and completing the square gives (x + 2)^2 = 7.
  14. 14

    Solve by completing the square: 3x^2 - 6x - 9 = 0.

    The solutions are x = 3 and x = -1. Dividing by 3 gives x^2 - 2x - 3 = 0, and completing the square gives (x - 1)^2 = 4.
  15. 15

    A ball's height is modeled by h(t) = t^2 - 6t + 13. Rewrite the function in vertex form and state the minimum height.

    Complete the square on t^2 - 6t.

    The function in vertex form is h(t) = (t - 3)^2 + 4. The minimum height is 4, which occurs at t = 3.
LivePhysics.com Math - Grade 9-12 - Answer Key