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

Verified step by step guidance

1. Recall that to find the square of a complex number, such as \((2 + 3i)^2\), you can use the formula for the square of a binomial: \((a + b)^2 = a^2 + 2ab + b^2\).
2. Identify \(a = 2\) and \(b = 3i\) in the expression \((2 + 3i)^2\).
3. Apply the formula: calculate \(a^2 = (2)^2\), \(2ab = 2 \times 2 \times 3i\), and \(b^2 = (3i)^2\) separately.
4. Remember that \(i^2 = -1\), so when you calculate \(b^2 = (3i)^2\), rewrite it as \(3^2 \times i^2 = 9 \times (-1)\).
5. Combine all the terms: \(a^2 + 2ab + b^2\), simplify the real and imaginary parts separately to write the result in standard form \(x + yi\).

Key Concepts

Complex Numbers and Standard Form

Complex numbers are expressed in the form a + bi, where a is the real part and b is the imaginary part. The standard form means writing the result explicitly as a sum of a real number and an imaginary number multiplied by i.

Binomial Expansion (Square of a Binomial)

Squaring a binomial (x + y)² involves expanding it as x² + 2xy + y². This method is essential for multiplying complex numbers like (2 + 3i)² by treating 2 and 3i as the terms x and y.

Properties of the Imaginary Unit i

The imaginary unit i satisfies i² = -1. This property is crucial when simplifying powers of i during multiplication, converting terms like (3i)² into real numbers to write the final answer in standard form.