In Exercises 9–20, find each product and write the result in standard form. (2 + 3i)^2

Verified step by step guidance

1. Identify the expression to be squared: \((2 + 3i)^2\).
2. Apply the formula for squaring a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).
3. Substitute \(a = 2\) and \(b = 3i\) into the formula: \((2)^2 + 2(2)(3i) + (3i)^2\).
4. Calculate each term separately: \(2^2\), \(2 \times 2 \times 3i\), and \((3i)^2\).
5. Combine the real and imaginary parts to write the result in standard form \(a + bi\).

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 express the product in standard form, a + bi.

Standard Form of Complex Numbers

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