Completing the square

The two places a quadratic goes past factoring: completing the square, whole numbers and then halves, and the roots that turn out to be complex.

4 sheets · 32 questions · one answer key · 5 pages

Recommended for grades 10–11

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Completing the square Sheet 1 of 4

Completing the squareUp to 12

Name: Date:

Write each expression as (x + p)² + q.

1x² − 20x + 104

2x² + 20x + 80

3x² − 18x + 96

4x² − 12x + 44

5x² + 6x + 11

6x² + 20x + 91

7x² + 8x + 22

8x² − 22x + 122

Completing the squareUp to 12

Name: Date:

Write each expression as (x + p)² + q. Some of these need halves and quarters.

1x² − 19x + 7

2x² + 3x − 9

3x² − 9x − 4

4x² + 19x − 11

5x² − 5x − 5

6x² + 23x − 2

7x² − 3x − 2

8x² + 23x + 11

Quadratics with complex solutionsPure imaginary roots

Name: Date:

Solve each equation. Every answer is imaginary.

1x² + 25 = 0

2x² + 9 = 0

3x² + 36 = 0

4x² + 1 = 0

5x² + 49 = 0

6x² + 64 = 0

7x² + 4 = 0

8x² + 16 = 0

Quadratics with complex solutionsComplex roots

Name: Date:

Solve each equation. Every answer is a complex number.

1x² − 8x + 32 = 0

2x² − 6x + 25 = 0

3x² + 8x + 25 = 0

4x² + 4x + 13 = 0

5x² + 10x + 26 = 0

6x² + 10x + 89 = 0

7x² − 10x + 61 = 0

8x² − 2x + 50 = 0

Power Pack · Completing the squareAnswer key

Completing the square

1. x² − 20x + 104 = (x − 10)² + 42. x² + 20x + 80 = (x + 10)² − 203. x² − 18x + 96 = (x − 9)² + 154. x² − 12x + 44 = (x − 6)² + 85. x² + 6x + 11 = (x + 3)² + 26. x² + 20x + 91 = (x + 10)² − 97. x² + 8x + 22 = (x + 4)² + 68. x² − 22x + 122 = (x − 11)² + 1

Halves and quarters

1. x² − 19x + 7 = (x − 19/2)² − 333/42. x² + 3x − 9 = (x + 3/2)² − 45/43. x² − 9x − 4 = (x − 9/2)² − 97/44. x² + 19x − 11 = (x + 19/2)² − 405/45. x² − 5x − 5 = (x − 5/2)² − 45/46. x² + 23x − 2 = (x + 23/2)² − 537/47. x² − 3x − 2 = (x − 3/2)² − 17/48. x² + 23x + 11 = (x + 23/2)² − 485/4

Pure imaginary roots

1. x² + 25 = 0 → x = ±5i2. x² + 9 = 0 → x = ±3i3. x² + 36 = 0 → x = ±6i4. x² + 1 = 0 → x = ±i5. x² + 49 = 0 → x = ±7i6. x² + 64 = 0 → x = ±8i7. x² + 4 = 0 → x = ±2i8. x² + 16 = 0 → x = ±4i

Complex pairs

1. x² − 8x + 32 = 0 → x = 4 ± 4i2. x² − 6x + 25 = 0 → x = 3 ± 4i3. x² + 8x + 25 = 0 → x = −4 ± 3i4. x² + 4x + 13 = 0 → x = −2 ± 3i5. x² + 10x + 26 = 0 → x = −5 ± i6. x² + 10x + 89 = 0 → x = −5 ± 8i7. x² − 10x + 61 = 0 → x = 5 ± 6i8. x² − 2x + 50 = 0 → x = 1 ± 7i

Completing the square

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