Solving Quadratic Equations Test Answer Key with Scoring – Algebra 1

Question 1 Square Roots (2 points)

(x + 3)² = 25, so x + 3 = ±5.

x = 2 or x = −8

Scoring: 1 point for isolating the square, 1 point for both solutions.

Question 2 Factoring (2 points)

1. (x + 4)(x + 7) = 0: x = −4 or x = −7.
2. 3x(x − 4) = 0: x = 0 or x = 4.

Scoring: 1 point per part; both solutions are needed.

Question 3 Complete the Square (3 points)

x² + 10x = −7. Add 25: (x + 5)² = 18.

x = −5 ± √18 = −5 ± 3√2 ≈ −5 ± 4.243.

x ≈ −0.76 or x ≈ −9.24

Scoring: 1 point for adding 25 to both sides, 1 point for the square root step, 1 point for both decimals.

Question 4 Quadratic Formula (3 points)

a = 2, b = −3, c = −7. Discriminant: 9 + 56 = 65.

x = 3 ± √654 ≈ 3 ± 8.0624.

x ≈ 2.77 or x ≈ −1.27

Scoring: 1 point for the discriminant, 1 point for the setup, 1 point for both decimals.

Question 5 Discriminant (2 points)

1. 36 − 48 = −12 < 0: no real solutions.
2. 36 − 36 = 0: exactly one real solution (x = −3).

Scoring: 1 point per part for the discriminant and the conclusion.

Question 6 Garden Dimensions (3 points)

w(w + 7) = 144, so w² + 7w − 144 = 0. Discriminant: 49 + 576 = 625, √625 = 25.

w = −7 ± 252: 9 or −16. Width cannot be negative.

Width 9 ft, length 16 ft (9 × 16 = 144).

Scoring: 1 point for the equation, 1 point for solving, 1 point for rejecting the negative width and giving both dimensions.

Question 7 When Does the Ball Land? (3 points)

1. −16t² + 24t + 16 = 0, or 2t² − 3t − 2 = 0 after dividing by −8.
2. (2t + 1)(t − 2) = 0, so t = −12 or t = 2.
3. Time cannot be negative, so the ball lands after 2 seconds.

Scoring: 1 point per part.

Question 8 Zero-Product Property (2 points)

If a product of two numbers is 0, at least one of the numbers is 0. So 2x − 5 = 0 or x + 8 = 0.

x = 52 or x = −8

Scoring: 1 point for the explanation, 1 point for both solutions.