Quadratic equations

Solving by square roots and by factoring, and deciding how many real solutions an equation has.

5 sheets · 50 questions · one answer key · 7 pages

Recommended for grades 9–10

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By square roots Sheet 1 of 5

Solve by square rootsSquare and shifted forms

Name: Date:

Solve each equation by taking square roots.

1(x − 5)² = 16

23x² = 243

3(x − 9)² = 25

4x² = 64

5(x + 9)² = 25

6(x − 3)² = 25

7x² = 100

88x² = 288

9x² = 9

10(x + 9)² = 100

Solving quadraticsAlready factored

Name: Date:

Each product is zero. Write the two values of x.

1(x + 8)(x − 14) = 0

2(x + 5)(x + 1) = 0

3(x − 2)(x − 12) = 0

4(x + 5)(x + 19) = 0

5(x − 20)(x − 16) = 0

6(x + 6)(x + 20) = 0

7(x + 15)(x − 20) = 0

8(x − 13)(x + 20) = 0

9(x + 2)(x − 1) = 0

10(x − 11)(x − 15) = 0

Solving quadraticsFactor it first

Name: Date:

Factor first, then solve. Write the two values of x.

1x² + 26x + 153 = 0

2x² − 12x + 35 = 0

3x² + 11x − 42 = 0

4x² + 19x + 84 = 0

5x² + 33x + 266 = 0

6x² + 5x − 14 = 0

7x² + 8x − 9 = 0

8x² − 18x + 77 = 0

9x² + 21x + 80 = 0

10x² − 23x + 102 = 0

Solving quadraticsNot equal to zero yet

Name: Date:

Make one side zero first, then solve.

1x² + 24x = −135

2x² − 20x = −75

3x² + 7x = 198

4x² − 9x = 112

5x² − 16x = −28

6x² + 2x = 323

7x² + 3x = 54

8x² + 12x = −32

9x² − 20x = −91

10x² + 21x = −68

How many real solutionsX² on its own

Name: Date:

How many real solutions does each equation have? Circle one.

1x² + 13x + 155 = 0

2x² − 15x + 87 = 0

3x² − 3x − 144 = 0

4x² − 27x − 119 = 0

5x² − 2x + 28 = 0

6x² − 33x + 6 = 0

7x² + 3x + 128 = 0

8x² − 14x + 49 = 0

9x² − 20x + 100 = 0

10x² + 10x + 25 = 0

Power Pack · Quadratic equationsAnswer key

By square roots

1. (x − 5)² = 16 → x = 1 or x = 92. 3x² = 243 → x = −9 or x = 93. (x − 9)² = 25 → x = 4 or x = 144. x² = 64 → x = −8 or x = 85. (x + 9)² = 25 → x = −14 or x = −46. (x − 3)² = 25 → x = −2 or x = 87. x² = 100 → x = −10 or x = 108. 8x² = 288 → x = −6 or x = 69. x² = 9 → x = −3 or x = 310. (x + 9)² = 100 → x = −19 or x = 1

Already factored

1. (x + 8)(x − 14) = 0 → (x + 8)(x − 14) = 0 → x = −8 or x = 142. (x + 5)(x + 1) = 0 → (x + 5)(x + 1) = 0 → x = −5 or x = −13. (x − 2)(x − 12) = 0 → (x − 2)(x − 12) = 0 → x = 2 or x = 124. (x + 5)(x + 19) = 0 → (x + 5)(x + 19) = 0 → x = −5 or x = −195. (x − 20)(x − 16) = 0 → (x − 20)(x − 16) = 0 → x = 20 or x = 166. (x + 6)(x + 20) = 0 → (x + 6)(x + 20) = 0 → x = −6 or x = −207. (x + 15)(x − 20) = 0 → (x + 15)(x − 20) = 0 → x = −15 or x = 208. (x − 13)(x + 20) = 0 → (x − 13)(x + 20) = 0 → x = 13 or x = −209. (x + 2)(x − 1) = 0 → (x + 2)(x − 1) = 0 → x = −2 or x = 110. (x − 11)(x − 15) = 0 → (x − 11)(x − 15) = 0 → x = 11 or x = 15

Standard form

1. x² + 26x + 153 = 0 → (x + 9)(x + 17) = 0 → x = −9 or x = −172. x² − 12x + 35 = 0 → (x − 5)(x − 7) = 0 → x = 5 or x = 73. x² + 11x − 42 = 0 → (x + 14)(x − 3) = 0 → x = −14 or x = 34. x² + 19x + 84 = 0 → (x + 12)(x + 7) = 0 → x = −12 or x = −75. x² + 33x + 266 = 0 → (x + 19)(x + 14) = 0 → x = −19 or x = −146. x² + 5x − 14 = 0 → (x − 2)(x + 7) = 0 → x = 2 or x = −77. x² + 8x − 9 = 0 → (x + 9)(x − 1) = 0 → x = −9 or x = 18. x² − 18x + 77 = 0 → (x − 7)(x − 11) = 0 → x = 7 or x = 119. x² + 21x + 80 = 0 → (x + 16)(x + 5) = 0 → x = −16 or x = −510. x² − 23x + 102 = 0 → (x − 17)(x − 6) = 0 → x = 17 or x = 6

Rearrange, then solve

1. x² + 24x = −135 → (x + 9)(x + 15) = 0 → x = −9 or x = −152. x² − 20x = −75 → (x − 15)(x − 5) = 0 → x = 15 or x = 53. x² + 7x = 198 → (x − 11)(x + 18) = 0 → x = 11 or x = −184. x² − 9x = 112 → (x − 16)(x + 7) = 0 → x = 16 or x = −7

Power Pack · Quadratic equationsAnswer key

Rearrange, then solve

5. x² − 16x = −28 → (x − 2)(x − 14) = 0 → x = 2 or x = 146. x² + 2x = 323 → (x − 17)(x + 19) = 0 → x = 17 or x = −197. x² + 3x = 54 → (x + 9)(x − 6) = 0 → x = −9 or x = 68. x² + 12x = −32 → (x + 4)(x + 8) = 0 → x = −4 or x = −89. x² − 20x = −91 → (x − 13)(x − 7) = 0 → x = 13 or x = 710. x² + 21x = −68 → (x + 17)(x + 4) = 0 → x = −17 or x = −4

Real solutions

1. x² + 13x + 155 = 0: 13² − 4 × 1 × 155 = 169 − 620 = −451 → None2. x² − 15x + 87 = 0: −15² − 4 × 1 × 87 = 225 − 348 = −123 → None3. x² − 3x − 144 = 0: −3² − 4 × 1 × −144 = 9 − -576 = 585 → Two4. x² − 27x − 119 = 0: −27² − 4 × 1 × −119 = 729 − -476 = 1205 → Two5. x² − 2x + 28 = 0: −2² − 4 × 1 × 28 = 4 − 112 = −108 → None6. x² − 33x + 6 = 0: −33² − 4 × 1 × 6 = 1089 − 24 = 1065 → Two7. x² + 3x + 128 = 0: 3² − 4 × 1 × 128 = 9 − 512 = −503 → None8. x² − 14x + 49 = 0: −14² − 4 × 1 × 49 = 196 − 196 = 0 → One9. x² − 20x + 100 = 0: −20² − 4 × 1 × 100 = 400 − 400 = 0 → One10. x² + 10x + 25 = 0: 10² − 4 × 1 × 25 = 100 − 100 = 0 → One

Quadratic equations

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