In Exercises 65–70, perform the indicated operation(s) and write the result in standard form. (2 + i)² - (3 - i)²

Verified step by step guidance

1. Step 1: Recall the formula for squaring a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Apply this formula to \((2 + i)^2\), where \(a = 2\) and \(b = i\). Expand it as \(2^2 + 2(2)(i) + i^2\).
2. Step 2: Simplify \((2 + i)^2\) by calculating each term: \(2^2 = 4\), \(2(2)(i) = 4i\), and \(i^2 = -1\) (since \(i^2 = -1\) by definition of the imaginary unit). Combine these to get \(4 + 4i - 1\).
3. Step 3: Simplify \(4 + 4i - 1\) to \(3 + 4i\). This is the result of \((2 + i)^2\).
4. Step 4: Similarly, apply the formula for squaring a binomial to \((3 - i)^2\), where \(a = 3\) and \(b = -i\). Expand it as \(3^2 + 2(3)(-i) + (-i)^2\).
5. Step 5: Simplify \((3 - i)^2\) by calculating each term: \(3^2 = 9\), \(2(3)(-i) = -6i\), and \((-i)^2 = -1\). Combine these to get \(9 - 6i - 1\), which simplifies to \(8 - 6i\). Subtract \((3 - i)^2\) from \((2 + i)^2\) to get \((3 + 4i) - (8 - 6i)\), and simplify the real and imaginary parts separately.

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 how to manipulate complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Standard Form of Complex Numbers

The standard form of a complex number is 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. This helps in further calculations and interpretations in complex number theory.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, such as (a + b)^n. In the context of complex numbers, this involves applying the formula (x + y)^2 = x^2 + 2xy + y^2 to simplify expressions like (2 + i)^2 and (3 - i)^2. Mastery of this concept is crucial for accurately calculating the squares of binomials.