Complex Solutions of Quadratics Test Answer Key with Scoring – Algebra 2

Question 1 Discriminants (2 points)

a) D = (−10)² − 4(1)(25) = 0: one repeated real solution.

b) D = (1)² − 4(2)(−15) = 121: two distinct real solutions.

c) D = (−2)² − 4(1)(8) = −28: two non-real complex solutions.

d) D = (4)² − 4(4)(5) = −64: two non-real complex solutions.

Scoring: 0.5 point per correct discriminant with its classification.

Question 2 Squared binomial equation (2 points)

x − 5 = ±√(−36) = ±6i.

x = 5 ± 6i

Scoring: 1 point for ±6i, 1 point for the final solutions.

Question 3 Quadratic formula (3 points)

Here a = 3, b = 2, c = 7.

Discriminant: D = b² − 4ac = (2)² − 4(3)(7) = −80.

D < 0, so there are two non-real solutions. √(−80) = 4i√5.

x = −(2) ± √(−80)2(3)

x = −1 ± 2i√53

Scoring: 1 point for the discriminant, 1 for simplifying √D, 1 for the final reduced solutions.

Question 4 Completing the square (3 points)

x² + 8x = −41. Add 16: (x + 4)² = −25.

x + 4 = ±5i.

x = −4 ± 5i

Scoring: 1 point for the perfect-square form, 1 for the square root step, 1 for the solutions.

Question 5 Range of a constant (2 points)

D = 36 − 4k < 0 gives k > 9.

k > 9

Scoring: 1 point for D, 1 point for the inequality.

Question 6 A graph and its equation (2 points)

a) The vertex (−1, 4) is above the x-axis and the parabola opens upward, so there are no real solutions.

b) D = 4 − 20 = −16, so x = −2 ± 4i2 = −1 ± 2i.

Scoring: 1 point each part.

Question 7 Writing an equation (3 points)

a) The conjugate: 2 − 7i.

b) Sum = 4; product = 4 + 49 = 53.

c) x² − 4x + 53 = 0

Scoring: 1 point each part.

Question 8 Will the drone reach the height? (3 points)

a) −16t² + 20t + 6 = 15 gives 16t² − 20t + 9 = 0.

b) D = (−20)² − 4(16)(9) = 400 − 576 = −176 < 0: the drone never reaches 15 feet (maximum 12.25 feet).

c) √(−176) = 4i√11, so t = 20 ± 4i√1132 = 5 ± i√118.

Scoring: 1 point each part.