In Exercises 1–10, perform the indicated operations and write the result in standard form. (3 − 4i)²

Verified step by step guidance

1. Identify the expression to be expanded: \((3 - 4i)^2\).
2. Use the formula for the square of a binomial: \((a - b)^2 = a^2 - 2ab + b^2\).
3. Substitute \(a = 3\) and \(b = 4i\) into the formula: \((3)^2 - 2(3)(4i) + (4i)^2\).
4. Calculate each term separately: \(3^2\), \(-2 \times 3 \times 4i\), and \((4i)^2\).
5. Combine the real and imaginary parts to express 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 involving them, such as addition, subtraction, multiplication, and division.

Squaring a Binomial

Squaring a binomial involves applying the formula (a - b)² = a² - 2ab + b². This formula is crucial for expanding expressions like (3 - 4i)², as it allows us to systematically calculate the square of the binomial by squaring each term and accounting for the product of the two terms.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where a and b are real numbers. When performing operations on complex numbers, it is important to express the final result in this form to clearly identify the real and imaginary components, facilitating further calculations and interpretations.