Complex Numbers Worksheet Answers

Struggling with your complex numbers worksheet? You’re not alone! Complex numbers can seem, well, complex, at first. Between the imaginary unit ‘i’ and the various operations like addition, subtraction, multiplication, and division, it’s easy to get lost. This post is designed to help you out by providing the answers to a common complex numbers worksheet, along with some quick explanations to reinforce your understanding. Remember, math is a process of learning and practicing, so don’t be discouraged if you don’t get it right away. Use these answers as a guide to identify areas where you need more practice and review the relevant concepts.

This section provides the solutions to a typical complex numbers worksheet. It’s crucial to not just copy these answers but to understand *how* they were derived. Try working through the problems yourself first, then use these solutions to check your work and pinpoint any errors.

Part 1: Simplifying Complex Numbers

These problems typically involve expressing complex numbers in the standard form a + bi, where ‘a’ is the real part and ‘b’ is the imaginary part. This often involves simplifying expressions with ‘i’, remembering that i² = -1.

Question 1: Simplify √-25

This question tests your understanding of the imaginary unit ‘i’. Remember that √-1 = i. Therefore, √-25 = √(25 * -1) = √25 * √-1 = 5i.

Answer: 5i
Question 2: Simplify √-48

Similar to the previous question, we can break this down: √-48 = √(16 * 3 * -1) = √16 * √3 * √-1 = 4√3 * i = 4i√3.

Answer: 4i√3
Question 3: Simplify i¹⁷

To simplify powers of ‘i’, we use the cyclical nature of i: i¹ = i, i² = -1, i³ = -i, i⁴ = 1. After that, the pattern repeats. Divide the exponent by 4 and find the remainder. The remainder tells you which value corresponds to that power of i. 17 divided by 4 is 4 with a remainder of 1. Therefore, i¹⁷ = i¹ = i.

Answer: i
Question 4: Simplify i³⁰

Following the same logic as above, 30 divided by 4 is 7 with a remainder of 2. Therefore, i³⁰ = i² = -1.

Answer: -1

Part 2: Operations with Complex Numbers

This section focuses on adding, subtracting, multiplying, and dividing complex numbers. Remember that when adding or subtracting, you combine like terms (real with real and imaginary with imaginary). Multiplication involves using the distributive property (FOIL method), and division often requires multiplying the numerator and denominator by the complex conjugate of the denominator.

Question 5: (3 + 2i) + (1 – 5i)

Combine the real parts and the imaginary parts: (3 + 1) + (2i – 5i) = 4 – 3i.

Answer: 4 – 3i
Question 6: (7 – i) – (4 + 3i)

Distribute the negative sign and then combine like terms: 7 – i – 4 – 3i = (7 – 4) + (-i – 3i) = 3 – 4i.

Answer: 3 – 4i
Question 7: (2 + i)(3 – 2i)

Use the FOIL method (First, Outer, Inner, Last): (2 * 3) + (2 * -2i) + (i * 3) + (i * -2i) = 6 – 4i + 3i – 2i². Remember that i² = -1, so -2i² = -2(-1) = 2. Therefore, 6 – 4i + 3i + 2 = 8 – i.

Answer: 8 – i
Question 8: (5 + i) / (2 – i)

Multiply the numerator and denominator by the complex conjugate of the denominator, which is (2 + i): [(5 + i) / (2 – i)] * [(2 + i) / (2 + i)] = (10 + 5i + 2i + i²) / (4 + 2i – 2i – i²) = (10 + 7i – 1) / (4 + 1) = (9 + 7i) / 5 = 9/5 + (7/5)i.

Answer: 9/5 + (7/5)i

Part 3: Solving Quadratic Equations with Complex Solutions

Sometimes, quadratic equations will have solutions that are complex numbers. This occurs when the discriminant (b² – 4ac) is negative. You’ll use the quadratic formula to find these solutions.

Question 9: Solve x² + 4x + 5 = 0

Use the quadratic formula: x = [-b ± √(b² – 4ac)] / 2a. In this case, a = 1, b = 4, and c = 5. Therefore, x = [-4 ± √(4² – 4 * 1 * 5)] / 2 * 1 = [-4 ± √(16 – 20)] / 2 = [-4 ± √-4] / 2 = [-4 ± 2i] / 2 = -2 ± i.

Answer: -2 + i, -2 – i
Question 10: Solve 2x² – 6x + 7 = 0

Again, use the quadratic formula: x = [-b ± √(b² – 4ac)] / 2a. Here, a = 2, b = -6, and c = 7. Therefore, x = [6 ± √((-6)² – 4 * 2 * 7)] / 2 * 2 = [6 ± √(36 – 56)] / 4 = [6 ± √-20] / 4 = [6 ± 2i√5] / 4 = 3/2 ± (√5/2)i.

Answer: 3/2 + (√5/2)i, 3/2 – (√5/2)i

Remember to practice consistently. Work through similar problems to solidify your understanding of complex numbers. If you are still struggling with a specific type of problem, consult your textbook, notes, or teacher for additional help. Good luck!

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