Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Complex Numbers

College Algebra

1. Equations & Inequalities

Complex Numbers

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1:34 minutes

Problem 63

Textbook Question

Find each product. Write answers in standard form. (-2-3i)(-2+3i)

Verified step by step guidance

Step 1: Distribute each term in the first complex number to each term in the second complex number. This means you will multiply -2 by -2, -2 by 3i, -3i by -2, and -3i by 3i.

Step 2: Simplify each of these products. Remember that i^2 = -1.

Step 3: Combine like terms. In this case, the terms with 'i' in them are like terms.

Step 4: Write the final answer in standard form for complex numbers, which is a + bi, where 'a' is the real part and 'bi' is the imaginary part.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i', which is defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Multiplication of Complex Numbers

To multiply complex numbers, you apply the distributive property (also known as the FOIL method for binomials). For example, when multiplying (a + bi)(c + di), you multiply each part: ac, adi, bci, and bdi^2. Remember that i^2 equals -1, which is crucial for simplifying the result into standard form.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' is the real part and 'b' is the imaginary part. When performing operations with complex numbers, the final answer should be simplified to this form, ensuring clarity and consistency in representation. This is important for further mathematical operations and interpretations.