Algebra 2 Complex Numbers: Adding, Subtracting and Multiplying Test Solutions and Scoring (PDF)

1. \(\sqrt{-64} = 8i\)
2. \(\sqrt{-75} = i\sqrt{25\cdot 3} = 5i\sqrt{3}\)

(a) \(8i\); (b) \(5i\sqrt{3}\)

Scoring: 1 point for each part.

1. \(15 = 4\cdot 3 + 3\), so \(i^{15} = i^{3} = -i\).
2. \(102 = 4\cdot 25 + 2\), so \(i^{102} = i^{2} = -1\).

(a) \(-i\); (b) \(-1\)

Scoring: 1 point for each part.

1. \((6 – 2) + (-3 + 8)i = 4 + 5i\)
2. \(4 + i – 9 + 5i = -5 + 6i\)

(a) \(4 + 5i\); (b) \(-5 + 6i\)

Scoring: 1 point for each part.

1. \(3 – 12i + 2i – 8i^{2} = 3 – 10i + 8 = 11 – 10i\)
2. \(4 – 49i^{2} = 4 + 49 = 53\)

(a) \(11 – 10i\); (b) \(53\)

Scoring: Part (a): 1 point for the four correct partial products, 1 point for replacing i² and combining. Part (b): 1 point.

A is \(-3 + 2i\) and B is \(2 – 4i\). Sum: \((-3 + 2) + (2 – 4)i = -1 – 2i\).

A = \(-3 + 2i\), B = \(2 – 4i\), A + B = \(-1 – 2i\)

Scoring: 1 point for both numbers, 1 point for the sum.

\(3x^{2} = -48\), so \(x^{2} = -16\).

\(x = \pm 4i\).

\(x = 4i\) or \(x = -4i\)

Scoring: 1 point for isolating x², 1 point for both complex solutions.

\(3i^{2} = -3\). \(i^{5} = i\), so \(-2i^{5} = -2i\). \((1 + i)^{2} = 1 + 2i – 1 = 2i\).

Sum: \(-3 – 2i + 2i = -3\).

\(-3 + 0i = -3\)

Scoring: 1 point for each of the three terms simplified correctly (the final sum is then determined).

\((3 – i)(2 + 5i) = 6 + 15i – 2i – 5i^{2}\).

\(-5i^{2} = 5\), so \(6 + 5 + 13i = 11 + 13i\).

\(11 + 13i\) volts

Scoring: 1 point for the correct distribution, 1 point for the final answer.

Real parts: \(x + 3 = 7\), so \(x = 4\).

Imaginary parts: \(2 – y = 5\), so \(y = -3\).

\(x = 4\), \(y = -3\)

Scoring: 1 point for each variable.