In Exercises 53–58, perform the indicated operation(s) and write the result in standard form.(2 + i)² − (3 − i)²

Verified step by step guidance

1. First, expand the expression \((2 + i)^2\) using the formula \((a + b)^2 = a^2 + 2ab + b^2\).
2. Calculate \((2 + i)^2\) by substituting \(a = 2\) and \(b = i\), resulting in \(2^2 + 2 \cdot 2 \cdot i + i^2\).
3. Next, expand the expression \((3 - i)^2\) using the formula \((a - b)^2 = a^2 - 2ab + b^2\).
4. Calculate \((3 - i)^2\) by substituting \(a = 3\) and \(b = i\), resulting in \(3^2 - 2 \cdot 3 \cdot i + i^2\).
5. Subtract the result of \((3 - i)^2\) from \((2 + i)^2\) and simplify the expression to write it 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, which are fundamental in solving problems involving them.

Operations on Complex Numbers

Operations on complex numbers include addition, subtraction, multiplication, and division. When performing these operations, it is important to apply the distributive property and combine like terms, particularly when dealing with the imaginary unit i. For example, when squaring a complex number, one must remember that i² equals -1, which affects the outcome of the operation.

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 b represents the imaginary part. Writing complex numbers in standard form is crucial for clarity and consistency, especially when performing operations and comparing different complex numbers.