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ACT Math Complex Numbers: 19 Practice Problems with Step-by-Step Explanations

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ACT Math Complex Numbers: 19 Practice Problems with Step-by-Step Explanations

Complex numbers on the ACT test whether you can treat i (defined as √−1, with i2=−1) as a variable that follows all the normal rules of algebra, with one extra simplification step whenever i2 shows up. Adding and subtracting complex numbers just means combining the real parts together and the imaginary parts together, keeping them separate the whole time — never combine a real number with an imaginary one. Multiplying two complex numbers uses standard distribution (FOIL), and then you simplify any i2 term to −1 as a final step.

Dividing by a complex number requires a specific trick: multiply both the numerator and denominator by the denominator's complex conjugate (flip the sign of its imaginary part). This eliminates the imaginary part from the denominator entirely, since a complex number times its conjugate always produces a real number: (a+bi)(a−bi) = a2+b2. This same conjugate trick shows up as its own question type — recognizing which number, when multiplied by a given complex number, produces a real result.

A few other tools round out this chapter: the magnitude (or absolute value) of a complex number a+bi is √(a2+b2) — the same distance formula you'd use for a point (a,b) in the coordinate plane. And powers of i cycle through a pattern of 4 (i, −1, −i, 1) that repeats, which is worth memorizing rather than recalculating each time. Work through the 19 problems below, then click to reveal each step-by-step explanation.

Three problems from the original set (19–21) reference specific points plotted on a complex-plane graph. Since the exact point positions weren't legible enough in the source images to solve reliably, they've been left out for now — send over the graph coordinates and they can be added.

On this page:

  1. Problem 1 — Subtracting complex numbers
  2. Problem 2 — Adding complex numbers
  3. Problem 3 — Adding complex numbers
  4. Problem 4 — Subtracting complex numbers
  5. Problem 5 — Subtracting complex numbers
  6. Problem 6 — Subtracting complex numbers
  7. Problem 7 — Scalar multiplication and subtraction
  8. Problem 8 — Simplifying i squared
  9. Problem 9 — Multiplying conjugates
  10. Problem 10 — Multiplying complex numbers
  11. Problem 11 — Dividing complex numbers
  12. Problem 12 — Magnitude of a complex number
  13. Problem 13 — Multiplying complex numbers
  14. Problem 14 — Multiplying complex numbers
  15. Problem 15 — Complex conjugate
  16. Problem 16 — Sum of imaginary radicals
  17. Problem 17 — Powers of i
  18. Problem 18 — Multiplying with a variable
  19. Problem 19 — Identifying the conjugate pattern

Practice Problems

Problem 1

Which of the following gives the correct answer when you subtract x from z?

x = −3+2i
z = 5+4i

  • A) −2+6i
  • B) 2+6i
  • C) 8+2i
  • D) 8+6i
Correct Answer: C

Subtracting x from z means z−x: (5+4i)−(−3+2i) = 5−(−3) + (4−2)i = 8+2i.

Problem 2

What is the sum of the complex numbers 8+2i and 3+3i?

  • A) 16
  • B) 16i
  • C) 11+5i
  • D) 13+3i
Correct Answer: C

Add the real parts and the imaginary parts separately: (8+3) + (2+3)i = 11+5i.

Problem 3

What is the sum of the complex numbers 2+7i and 6+5i?

  • A) 8+12i
  • B) 20i
  • C) 9+11i
  • D) 12+35i
Correct Answer: A

Add the real parts and the imaginary parts separately: (2+6) + (7+5)i = 8+12i.

Problem 4

Which of the following complex numbers is equal to (3+6i) − (−5+3i)?

  • A) −8+3i
  • B) −2+3i
  • C) −2+9i
  • D) 8+3i
Correct Answer: D

Distribute the subtraction: 3+6i+5−3i.

Combine like terms: (3+5) + (6−3)i = 8+3i.

Problem 5

(−4+6i) − (1+3i) = ?

  • A) −5+9i
  • B) 5+3i
  • C) −5+3i
  • D) −3+91
Correct Answer: C

Subtract the real parts and the imaginary parts separately: (−4−1) + (6−3)i = −5+3i.

Problem 6

What is the difference between complex numbers (6+5i) and (2i+3)?

  • A) 8+2i
  • B) 3+3i
  • C) 3+7i
  • D) 9+3i
Correct Answer: B

Rewrite the second number in standard form: 2i+3 = 3+2i. Subtract: (6+5i)−(3+2i) = (6−3) + (5−2)i = 3+3i.

Problem 7

4(3+2i) − 0.5(−8+6i) = ?

  • A) 8+8i
  • B) 8+5i
  • C) 16+5i
  • D) 12+8i
Correct Answer: C

Distribute each scalar: 4(3+2i) = 12+8i, and 0.5(−8+6i) = −4+3i.

Subtract: (12+8i) − (−4+3i) = (12+4) + (8−3)i = 16+5i.

Problem 8

Which of the following complex numbers is equal to (5+4i) − (8i2−8i)?

  • A) −13−4i
  • B) −3−4i
  • C) 13−4i
  • D) 13+12i
Correct Answer: D

Simplify i2=−1 first: 8i2−8i = 8(−1)−8i = −8−8i.

Subtract: (5+4i) − (−8−8i) = (5+8) + (4+8)i = 13+12i.

Problem 9

What is the product of the complex numbers (3+4i) and (3−4i)?

  • A) 25
  • B) 7
  • C) 1
  • D) 9−16i
Correct Answer: A

These are complex conjugates, so use the pattern (a+bi)(a−bi) = a2+b2: 32+42 = 9+16 = 25.

Problem 10

What is the product of the complex numbers (4+5i) and (−3+3i)?

  • A) 27+3i
  • B) −27−3i
  • C) 2+11i
  • D) −27+11i
Correct Answer: B

Distribute (FOIL): 4(−3)+4(3i)+5i(−3)+5i(3i) = −12+12i−15i+15i2.

Simplify i2=−1: −12+12i−15i−15 = (−12−15) + (12−15)i = −27−3i.

Problem 11

In the complex numbers, where i2=−1, 4/(1−3i) = ?

  • A) −1/2 − 3/2 i
  • B) 2/5 + 6/5 i
  • C) 2/5 − 6/5 i
  • D) 4/5 − 12/5 i
Correct Answer: B

Multiply the numerator and denominator by the conjugate of the denominator, 1+3i: [4(1+3i)] / [(1−3i)(1+3i)].

The denominator becomes 12+32=10. The numerator becomes 4+12i. Result: (4+12i)/10 = 2/5 + 6/5 i.

Problem 12

|−6i+2| = ?

  • A) −4
  • B) 4
  • C) 4√2
  • D) 2√10
Correct Answer: D

Rewrite in standard form: −6i+2 = 2−6i. The magnitude of a+bi is √(a2+b2).

√(22+(−6)2) = √(4+36) = √40 = √(4×10) = 2√10.

Problem 13

What is the product of the complex numbers (6+3i) and (2+i)?

  • A) 8+4i
  • B) 15+12i
  • C) 9+12i
  • D) 12+9i
Correct Answer: C

Distribute (FOIL): 6(2)+6(i)+3i(2)+3i(i) = 12+6i+6i+3i2.

Simplify i2=−1: 12+12i−3 = 9+12i.

Problem 14

For i = √−1, (5−5i)(−5+5i) = ?

  • A) −25−25i
  • B) −25−50i
  • C) 25i
  • D) 50i
Correct Answer: D

Distribute (FOIL): 5(−5)+5(5i)+(−5i)(−5)+(−5i)(5i) = −25+25i+25i−25i2.

Simplify i2=−1: −25+50i+25 = 0+50i = 50i.

Problem 15

Which of the following is the complex conjugate of −4+9i?

  • A) 4+9i
  • B) −4−9i
  • C) √97
  • D) 5i
Correct Answer: B

The complex conjugate of a+bi is a−bi — the real part stays the same, and the sign of the imaginary part flips.

The conjugate of −4+9i is −4−9i.

Problem 16

What is the sum of complex numbers √−48 and √−75?

  • A) i√123
  • B) 9i√3
  • C) 20i√3
  • D) 60i
Correct Answer: B

Rewrite each as an imaginary number: √−48 = i√48 = i√(16×3) = 4i√3. √−75 = i√75 = i√(25×3) = 5i√3.

Add: 4i√3 + 5i√3 = 9i√3.

Problem 17

If i = √−1, what is the value of the expression 12i4+8i2+12?

  • A) 32
  • B) 24
  • C) 16
  • D) 8
Correct Answer: C

Simplify each power of i: i4=1, and i2=−1.

Substitute: 12(1)+8(−1)+12 = 12−8+12 = 16.

Problem 18

For all real numbers x and imaginary number i, which of the following expressions is equivalent to (x+4i)(2x−5i)?

  • A) 2x2−13xi+20
  • B) 2x2+3xi−20
  • C) 2x2+13xi−20
  • D) 2x2+3xi+20
Correct Answer: D

Distribute (FOIL): x(2x)+x(−5i)+4i(2x)+4i(−5i) = 2x2−5xi+8xi−20i2.

Simplify i2=−1: 2x2+3xi+20.

Problem 19

The product of a+bi and which of the following complex numbers results in a real number?

  • A) b+ai
  • B) abi
  • C) a−bi
  • D) b−ai
Correct Answer: C

Multiplying a complex number by its conjugate always produces a real number: (a+bi)(a−bi) = a2+b2.

Testing the other options shows they leave a nonzero imaginary term in general, so only the conjugate a−bi reliably gives a real result.

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