Algebra 2 Practice Test: Complex Numbers and Quadratics
- Complex Numbers and Quadratics
- 15 questions
- 20 minutes
These 15 questions practice powers of i, adding, multiplying and dividing complex numbers, square roots of negative numbers, quadratic equations with complex solutions, the discriminant, vertex form and quadratic inequalities. Remember that i2 = −1, so replace every i2 by −1 before you collect terms.
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Question 1
What is the value of i23?
Correct answer: A
Powers of i repeat every 4 steps. 23 = 4 · 5 + 3, so i23 = i3 = −i.
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Question 2
Which expression is equivalent to (3 + 2i)(4 − 5i)?
Correct answer: B
12 − 15i + 8i − 10i2 = 12 − 7i + 10 = 22 − 7i. Treating −10i2 as −10 gives 2 − 7i.
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Question 3
Which expression is equivalent to 5 + i2 − i?
Correct answer: D
Multiply numerator and denominator by the conjugate 2 + i: (5 + i)(2 + i) = 10 + 7i + i2 = 9 + 7i, and (2 − i)(2 + i) = 4 + 1 = 5. The denominator 3 comes from computing 4 − 1.
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Question 4
What are the solutions of x2 + 2x + 10 = 0?
Correct answer: A
x = −2 ± 4 − 402 = −2 ± −362 = −2 ± 6i2 = −1 ± 3i. Not dividing the 6i by 2 gives −2 ± 6i.
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Question 5
Which expression is equal to −48?
Correct answer: A
−48 = −1 · 16 · 3 = 4i3. The square root of a negative number is imaginary, so −43, a real number, is wrong.
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Question 6
Simplify (6 − 4i) − (−2 + 3i).
Correct answer: D
Subtract real and imaginary parts separately: 6 − (−2) = 8 and −4i − 3i = −7i. Forgetting to distribute the minus sign gives 4 − i.
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Question 7
Which equation has two non-real complex solutions?
Correct answer: B
For x2 − 4x + 7 = 0 the discriminant is 16 − 28 = −12 < 0, so the solutions are complex. The discriminants of the others are 44, 0 and 4, giving real solutions.
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Question 8
What is the product (7 − 2i)(7 + 2i)?
Correct answer: C
(a − bi)(a + bi) = a2 + b2 = 49 + 4 = 53. Using −4i2 = −4 instead of +4 gives 45.
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Question 9
Which quadratic equation has the solutions 2 + i and 2 − i?
Correct answer: C
The sum of the solutions is 4 and the product is (2 + i)(2 − i) = 4 + 1 = 5, so the equation is x2 − 4x + 5 = 0. A product of 3 would come from 4 − 1.
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Question 10
Which is the vertex form of y = 2x2 − 12x + 13?
Correct answer: A
2(x2 − 6x) + 13 = 2(x2 − 6x + 9) − 18 + 13 = 2(x − 3)2 − 5. Subtracting 9 instead of 2 · 9 = 18 gives +4.
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Question 11
What is the solution set of x2 − x − 12 < 0?
Correct answer: B
(x − 4)(x + 3) < 0 when the factors have opposite signs, that is between the zeros: −3 < x < 4. The "or" form is the solution of x2 − x − 12 > 0.
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Question 12
What is the absolute value (modulus) of 6 − 8i?
Correct answer: C
|a + bi| = a2 + b2 = 36 + 64 = 100 = 10. The value 100 is the result before taking the square root.
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Question 13
For which positive value of k does x2 + kx + 16 = 0 have exactly one real solution?
Correct answer: B
Exactly one real solution means the discriminant is 0: k2 − 64 = 0, so k = 8 (or −8). Then x2 + 8x + 16 = (x + 4)2.
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Question 14
What are the solutions of 3(x − 2)2 + 12 = 0?
Correct answer: C
3(x − 2)2 = −12, so (x − 2)2 = −4 and x − 2 = ±2i. Then x = 2 ± 2i. Dropping the i gives the real numbers 0 and 4, which do not satisfy the equation.
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Question 15
A store's monthly profit, in dollars, is P(x) = −5x2 + 200x − 1200, where x is the price of an item in dollars. What is the maximum monthly profit?
Correct answer: D
The maximum is at x = −2002(−5) = 20, and P(20) = −2000 + 4000 − 1200 = 800. $20 is the best price, not the profit.
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