Find each product. Write answers in standard form. (-2-3i)(-2+3i)

Verified step by step guidance

1. Recognize that the expression is a product of two complex conjugates: \((-2 - 3i)\) and \((-2 + 3i)\).
2. Recall the formula for the product of conjugates: \((a - bi)(a + bi) = a^2 + b^2\), where \(a\) and \(b\) are real numbers.
3. Identify \(a = -2\) and \(b = 3\) from the given complex numbers.
4. Calculate \(a^2\) and \(b^2\): compute \((-2)^2\) and \$3^2$ separately.
5. Add the results from the previous step to get the product in standard form: \(a^2 + b^2\).

Key Concepts

Complex Numbers and Imaginary Unit

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. Understanding this form is essential for performing operations like addition, subtraction, and multiplication involving complex numbers.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property (FOIL method) to expand the product, then simplifying by combining like terms and applying i² = -1. This process transforms the product into a standard form a + bi.

Standard Form of a Complex Number

The standard form of a complex number is expressed as a + bi, where a is the real part and b is the imaginary coefficient. Writing answers in this form ensures clarity and consistency in representing complex numbers after operations.