In Exercises 9–20, find each product and write the result in standard form. (5 - 2i)²

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1. Recognize that the expression (5 - 2i)^2 is a binomial squared. Use the formula for the square of a binomial: (a - b)^2 = a^2 - 2ab + b^2.
2. Identify the values of a and b in the binomial. Here, a = 5 and b = 2i.
3. Substitute the values of a and b into the formula: (5 - 2i)^2 = 5^2 - 2(5)(2i) + (2i)^2.
4. Simplify each term: 5^2 = 25, -2(5)(2i) = -20i, and (2i)^2 = 4i^2. Recall that i^2 = -1, so 4i^2 = 4(-1) = -4.
5. Combine the simplified terms: 25 - 20i - 4. Write the result in standard form as a + bi, where a is the real part and b is the imaginary part.

Key Concepts

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, b is the imaginary part, and i is the imaginary unit 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) and combine like terms. When multiplying, remember that i^2 equals -1, which is crucial for simplifying the result. This process allows you to find the product of two complex numbers effectively.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where a and b are real numbers. In this form, a represents the real part, and bi represents the imaginary part. Writing complex numbers in standard form is important for clarity and consistency in mathematical communication, especially when performing further operations.