ACT Math Quiz: Complex Numbers

What this quiz covers

This quiz focuses on Complex Numbers, giving you a quick way to practice the rules, question types, and explanations that matter most for ACT Math.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $1 + 2i$ (correct answer)
2. $2 + i$
3. $1 - 2i$
4. $2 - i$

Explanation: The correct answer is A (1 + 2i). Multiply by the conjugate of the denominator: (3 + i)/(1 − i) × (1 + i)/(1 + i). Numerator: (3 + i)(1 + i) = 3 + 3i + i + i² = 3 + 4i − 1 = 2 + 4i. Denominator: (1 − i)(1 + i) = 1 − i² = 1 + 1 = 2. Result: (2 + 4i)/2 = 1 + 2i. B (2 + i) likely comes from incomplete multiplication or forgetting to divide. C (1 − 2i) comes from a sign error in the numerator expansion. D (2 − i) comes from multiplying by the wrong conjugate (1 − i) instead of (1 + i). The key technique: always multiply by the conjugate to eliminate i from the denominator, remembering that i² = −1.

Question 2

1. $6+7i$
2. $-6-7i$ (correct answer)
3. $6-7i$
4. $-6+7i$

Explanation: This problem asks for the complex conjugate of $-6 + 7i$, which is found by changing the sign of the imaginary part only. The real part is $-6$, and the imaginary part $7i$ becomes $-7i$. Thus, the conjugate is $-6 - 7i$. Choice D might result from incorrectly flipping the sign of the real part instead of the imaginary part.

Question 3

1. $3 + 3i$
2. $3 + 7i$ (correct answer)
3. $5 + 3i$
4. $5 + 7i$

Explanation: This is a complex numbers question testing subtraction with distribution. Choice B (3 + 7i) is correct — distribute the negative: (4 + 5i) − (1 − 2i) = 4 + 5i − 1 + 2i. Key step: −(−2i) = +2i. Combine real parts: 4 − 1 = 3. Combine imaginary parts: 5i + 2i = 7i. Result: 3 + 7i. Choice A (3 + 3i) correctly subtracts the real parts but fails to distribute the negative on the imaginary term: 5i − 2i = 3i instead of 5i + 2i = 7i. Choice C (5 + 3i) adds the real parts instead of subtracting: 4 + 1 = 5, and also gets the imaginary term wrong. Choice D (5 + 7i) adds real parts (correctly gets +7i from the imaginary) — two separate errors that partially cancel. Pro tip: When subtracting a complex number, rewrite it as addition of the negative first: (4 + 5i) + (−1 + 2i). This prevents sign errors by making every operation an addition. The most common mistake is treating −(−2i) as −2i instead of +2i.

Question 4

1. $-2 - 2i$
2. $-2 + 6i$ (correct answer)
3. $8 - 2i$
4. $-2 - 6i$

Explanation: This is a complex numbers question testing subtraction with distribution. Choice B (−2 + 6i) is correct — distribute the negative sign: (3 + 2i) − (5 − 4i) = 3 + 2i − 5 + 4i. The critical step: −(−4i) = +4i. Combine real parts: 3 − 5 = −2. Combine imaginary parts: 2i + 4i = 6i. Result: −2 + 6i. Choice A (−2 − 2i) gets the real part right but subtracts the imaginary parts without distributing the negative: treating it as 2i − 4i = −2i instead of 2i + 4i = 6i. Choice C (8 − 2i) adds the real parts instead of subtracting: 3 + 5 = 8, and also handles the imaginary term incorrectly. Choice D (−2 − 6i) gets the real part right but applies the sign error in the opposite direction — treating −(−4i) as −6i. Pro tip: When subtracting a complex number, rewrite the subtraction as adding the negative first: (3 + 2i) − (5 − 4i) becomes (3 + 2i) + (−5 + 4i). Then combine real and imaginary parts separately.

Question 5

1. 5 + 8i (correct answer)
2. 5 - 2i
3. 9 + 8i
4. 9 - 2i

Explanation: This problem requires subtracting two complex numbers. To subtract complex numbers, we subtract the real parts and subtract the imaginary parts. For (7 + 3i) - (2 - 5i), we get (7 - 2) + (3i - (-5i)) = 5 + (3i + 5i) = 5 + 8i. When subtracting a negative imaginary term, it becomes addition.

Question 6

1. $11 - 5i$
2. $5 - 5i$
3. $5 + 13i$ (correct answer)
4. $-5 + 13i$

Explanation: The operation is subtraction of two complex numbers: (8 + 4i) minus (3 - 9i). Distribute the negative: 8 + 4i - 3 + 9i. Combine real parts: 8 - 3 = 5. Combine imaginary parts: 4i + 9i = 13i. The result in standard form is 5 + 13i. Choice B might result from subtracting imaginary parts incorrectly.

Question 7

1. 5 - i
2. 5 + 5i
3. 3 - i
4. 3 + 5i (correct answer)

Explanation: This is subtraction of complex numbers, where we subtract corresponding parts. $(4 + 2i) - (1 - 3i) = (4 - 1) + (2i - (-3i)) = 3 + (2i + 3i) = 3 + 5i$. We subtract the real parts and subtract the imaginary parts separately.

Question 8

1. $3$
2. $\sqrt{63}$
3. $21$
4. $15$ (correct answer)

Explanation: The magnitude formula gives $|a + bi| = \sqrt{a^2 + b^2}$. For $(-9 + 12i)$, we have $a = -9$ and $b = 12$, so the magnitude is $\sqrt{(-9)^2 + 12^2} = \sqrt{81 + 144} = \sqrt{225} = 15$. The absolute value represents the distance from the origin to the point $(-9, 12)$ in the complex plane.

Question 9

1. 11 - 2i
2. 5 + 10i
3. -5 - 10i
4. -5 + 10i (correct answer)

Explanation: This is multiplication of complex numbers using FOIL. $(1 + 2i)(3 + 4i) = 1 \cdot 3 + 1 \cdot 4i + 2i \cdot 3 + 2i \cdot 4i = 3 + 4i + 6i + 8i^2$. Since $i^2 = -1$, this becomes $3 + 10i + 8(-1) = 3 + 10i - 8 = -5 + 10i$.

Question 10

1. 3 - 7i
2. -3 - 7i (correct answer)
3. -3 + 7i
4. 3 + 7i

Explanation: The complex conjugate of a complex number changes the sign of the imaginary part. For a complex number a + bi, the conjugate is a - bi. The complex conjugate of -3 + 7i is -3 - 7i. We keep the real part the same and change the sign of the imaginary part from positive to negative.

Question 11

1. 2
2. 14
3. 100
4. 10 (correct answer)

Explanation: The absolute value (magnitude) of a complex number $a + bi$ equals $\sqrt{a^2 + b^2}$. For $6 - 8i$, we calculate $\sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$. The absolute value is $10$.

Question 12

1. $2 - 7i$
2. $22 - 7i$ (correct answer)
3. $22 - 23i$
4. $12 - 15i + 10i^2$

Explanation: The operation is multiplication of two complex numbers, (3 + 2i) and (4 - 5i). Using FOIL, expand as follows: first terms 34=12, outer terms 3(-5i)=-15i, inner terms 2i4=8i, and last terms 2i(-5i)=-10i². Since i squared equals negative one, replace -10i² with -10*(-1)=+10. Combine the real parts 12 + 10 = 22 and the imaginary parts -15i + 8i = -7i, resulting in the standard form 22 - 7i. A common error, like in choice C, might involve mishandling the signs in the imaginary terms.

Question 13

1. 25
2. 0
3. -5
4. 5 (correct answer)

Explanation: The absolute value (magnitude) of a complex number $a + bi$ equals $\sqrt{a^2 + b^2}$. For the purely imaginary number $0 + 5i$, we calculate $\sqrt{0^2 + 5^2} = \sqrt{0 + 25} = \sqrt{25} = 5$. The absolute value is $5$.

Question 14

1. $2 \pm 3i$ (correct answer)
2. $-2 \pm 3i$
3. $4 \pm 6i$
4. The function has no complex roots

Explanation: This is a complex roots question testing the quadratic formula with a negative discriminant. Choice A (2 ± 3i) is correct — apply the quadratic formula: x = (4 ± √(16 − 52)) / 2 = (4 ± √(−36)) / 2 = (4 ± 6i) / 2 = 2 ± 3i. The discriminant is 16 − 52 = −36, which is negative, confirming complex (non-real) roots. Choice B (−2 ± 3i) correctly computes √(−36) = 6i and divides by 2, but uses −b instead of b/2a for the real part: applying −(−4)/2 = −2 rather than 4/2 = 2. Choice C (4 ± 6i) correctly computes the numerator 4 ± 6i but forgets to divide by 2 (the denominator of the quadratic formula). Choice D states "no complex roots" — this contradicts the fact that a negative discriminant guarantees two complex conjugate roots. Pro tip: A negative discriminant (b² − 4ac < 0) means the quadratic has complex roots, not "no roots." Write √(−36) = √36 · √(−1) = 6i, then divide the entire numerator by 2a.

Question 15

1. $-2 - 2i$
2. $-2 + 6i$ (correct answer)
3. $8 - 2i$
4. $-2 - 6i$

Explanation: This is a complex numbers question testing subtraction with distribution. Choice B (−2 + 6i) is correct — distribute the negative sign: (3 + 2i) − (5 − 4i) = 3 + 2i − 5 + 4i. The critical step: −(−4i) = +4i. Combine real parts: 3 − 5 = −2. Combine imaginary parts: 2i + 4i = 6i. Result: −2 + 6i. Choice A (−2 − 2i) gets the real part right but subtracts the imaginary parts without distributing the negative: treating it as 2i − 4i = −2i instead of 2i + 4i = 6i. Choice C (8 − 2i) adds the real parts instead of subtracting: 3 + 5 = 8, and also handles the imaginary term incorrectly. Choice D (−2 − 6i) gets the real part right but applies the sign error in the opposite direction — treating −(−4i) as −6i. Pro tip: When subtracting a complex number, rewrite the subtraction as adding the negative first: (3 + 2i) − (5 − 4i) becomes (3 + 2i) + (−5 + 4i). Then combine real and imaginary parts separately.

Question 16

1. -2 + 2i (correct answer)
2. 2 + 2i
3. -2 + 4i
4. 2 - 4i

Explanation: This is subtraction of complex numbers, where we subtract corresponding parts. $(2 + 3i) - (4 + i) = (2 - 4) + (3i - i) = -2 + 2i$. We subtract the real parts and subtract the imaginary parts separately.

Question 17

1. $-8+20i$
2. $-12-16i$
3. $12+16i$ (correct answer)
4. $12+16i^2$

Explanation: Using FOIL to multiply: $(-6 + 2i)(-1 - 3i) = (-6)(-1) + (-6)(-3i) + (2i)(-1) + (2i)(-3i) = 6 + 18i - 2i - 6i^2$. Since $i^2 = -1$, this becomes $6 + 18i - 2i - 6(-1) = 6 + 16i + 6 = 12 + 16i$. Choice D would result from not applying the rule that $i^2 = -1$.

Question 18

1. $2+11i$
2. $-2+11i$ (correct answer)
3. $-2-11i$
4. $2-11i$

Explanation: The complex conjugate changes the sign of the imaginary part while keeping the real part the same. For the complex number $-2 - 11i$, the real part is $-2$ and the imaginary part is $-11i$. The complex conjugate is $-2 - (-11i) = -2 + 11i$. This conjugate is essential for rationalizing denominators containing complex numbers.

Question 19

1. $-9-12i$ (correct answer)
2. $13-6i$
3. $-13-6i$
4. $9+12i$

Explanation: Subtracting complex numbers requires distributing the negative sign and combining like terms. $(2 - 9i) - (11 + 3i) = 2 - 9i - 11 - 3i$. Combining real parts: $2 - 11 = -9$, and combining imaginary parts: $-9i - 3i = -12i$. The result in standard form is $-9 - 12i$.

Question 20

1. 576
2. 31
3. 17
4. 25 (correct answer)

Explanation: The absolute value (or modulus) of a complex number $a + bi$ equals the square root of ($a^2 + b^2$). For $7 - 24i$, we calculate $|7 - 24i| = \sqrt{7^2 + (-24)^2} = \sqrt{49 + 576} = \sqrt{625} = 25$. This is a Pythagorean triple ($7, 24, 25$), making the calculation straightforward.