Solving Quadratic Equations Worksheet Answer Key – Algebra 1

Problem 1 Square Roots, Plain Case

Take both square roots: x = ±√81.

x = 9 or x = −9

Problem 2 Square Roots After Dividing

Divide by 3: x² = 25. Then x = ±5.

x = 5 or x = −5

Problem 3 Square Roots with a Shifted Square

Take square roots: x − 4 = 7 or x − 4 = −7.

Add 4: 11 or −3.

x = 11 or x = −3

Problem 4 Zero-Product Property

Either x + 6 = 0 or x − 2 = 0.

x = −6 or x = 2

Problem 5 Factor and Solve

Factor: (x + 2)(x + 5) = 0.

x = −2 or x = −5

Problem 6 No Constant Term

Factor out x: x(x − 9) = 0.

Either x = 0 or x − 9 = 0. Do not divide by x, because that would lose the solution 0.

x = 0 or x = 9

Problem 7 Factor a Trinomial Equation

Product −30 and sum −7: −10 and 3. So (x − 10)(x + 3) = 0.

x = 10 or x = −3

Problem 8 Factor with a Leading 2

ac = −20 with sum 1: 5 and −4. Split: 2x² − 4x + 5x − 10 = 2x(x − 2) + 5(x − 2) = (2x + 5)(x − 2).

2x + 5 = 0 gives x = −52.

x = −52 or x = 2

Problem 9 Irrational Square Roots

x + 2 = ±√18 = ±3√2. So x = −2 ± 3√2.

3√2 ≈ 4.243, so x ≈ 2.24 or x ≈ −6.24.

x = −2 ± 3√2 (about 2.24 and −6.24)

Problem 10 Complete the Square, Positive Middle

x² + 8x = 5. Half of 8 is 4, so add 16: x² + 8x + 16 = 21.

(x + 4)² = 21, so x = −4 ± √21 ≈ −4 ± 4.583.

x ≈ 0.58 or x ≈ −8.58

Problem 11 Complete the Square, Negative Middle

x² − 6x = −2. Add 9: (x − 3)² = 7.

x = 3 ± √7 ≈ 3 ± 2.646.

x ≈ 5.65 or x ≈ 0.35

Problem 12 Complete the Square, Leading 4

Divide by 4: x² + 2x = 14. Add 1: (x + 1)² = 54.

x + 1 = ±√52, so x = −1 ± √52 ≈ −1 ± 1.118.

x = −1 ± √52 (about 0.12 and −2.12)

Problem 13 Quadratic Formula, First Try

a = 1, b = 3, c = −7. The discriminant is 9 + 28 = 37.

x = −3 ± √372 ≈ −3 ± 6.0832.

x ≈ 1.54 or x ≈ −4.54

Problem 14 Count the Solutions

1. 36 − 36 = 0: one real solution (x = 3).
2. 9 − 40 = −31: no real solution.
3. 1 + 4 = 5: two real solutions.

(a) one; (b) none; (c) two

Problem 15 Formula with a = 3

a = 3, b = −5, c = −4. Discriminant: 25 + 48 = 73.

x = 5 ± √736 ≈ 5 ± 8.5446.

x ≈ 2.26 or x ≈ −0.59

Problem 16 Pick the Fast Method

Square roots are quickest because the equation has a squared binomial. Divide by 4: (x − 1)² = 25.

x − 1 = ±5, so x = 6 or x = −4.

x = 6 or x = −4 (square roots)

Problem 17 Rectangular Plot

Let the width be w. Then w(w + 3) = 154, so w² + 3w − 154 = 0.

Factor: (w + 14)(w − 11) = 0. Width cannot be −14.

Width 11 ft, length 14 ft. Check: 11 × 14 = 154 ✓.

11 ft by 14 ft

Problem 18 Height Equal to 30 Feet

Set −16t² + 40t + 6 = 30, so −16t² + 40t − 24 = 0. Divide by −8: 2t² − 5t + 3 = 0.

Factor: (2t − 3)(t − 1) = 0, so t = 1 or t = 32.

At 1 second (rising) and 1.5 seconds (falling)

Problem 19 Landing Time with the Formula

Set −16t² + 30t + 8 = 0 and multiply by −1: 16t² − 30t − 8 = 0, then divide by 2: 8t² − 15t − 4 = 0.

Discriminant: 225 + 128 = 353. t = 15 ± √35316 ≈ 15 ± 18.78816.

The minus case is negative, so reject it. t ≈ 33.788 ÷ 16 ≈ 2.11.

About 2.11 seconds

Problem 20 Exactly One Solution

Exactly one solution needs discriminant 0: 36 − 4k = 0, so k = 9.

None needs a negative discriminant: 36 − 4k < 0, so k > 9.

k = 9 for one solution; k > 9 for none