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. Recall that the standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with the property \(i^2 = -1\).
2. Start by expanding each square using the formula for the square of a binomial: \((x + y)^2 = x^2 + 2xy + y^2\). For \((2 + i)^2\), let \(x = 2\) and \(y = i\).
3. Similarly, expand \((3 - i)^2\) using the same formula, where \(x = 3\) and \(y = -i\).
4. After expanding both expressions, subtract the second result from the first, combining like terms (real parts together and imaginary parts together).
5. Simplify the expression by replacing \(i^2\) with \(-1\) and combining all real and imaginary terms to write the final answer in the form \(a + bi\).

Key Concepts

Complex Number Standard Form

A complex number is expressed in standard form as a + bi, where a is the real part and b is the imaginary part. Writing results in this form helps clearly separate the real and imaginary components, making it easier to interpret and use in further calculations.

Operations with Complex Numbers

Complex numbers can be added, subtracted, multiplied, and divided using algebraic rules, treating i as the imaginary unit with the property i² = -1. When performing operations like squaring or subtraction, apply distributive and associative properties carefully.

Squaring Complex Numbers

To square a complex number (a + bi), use the formula (a + bi)² = a² + 2abi + (bi)², remembering that i² = -1. This expands to (a² - b²) + 2ab i, which separates the result into real and imaginary parts for standard form.