Problem 80 - Algebra for College Students

Question

Find each of the products and express the answers in the standard form of a complex number. $$(-3-6 i)^{2}$$

Step-by-Step Solution

Answer

The product is \(-27 + 36i\).

Step 1: Identify the formula to expand

To find \[(-3-6i)^2\]we use the square of a binomial formula, which is:\[(a-b)^2 = a^2 - 2ab + b^2\] Here, identify \(a = -3\) and \(b = 6i\).

Step 2: Apply the formula

Substitute \(a = -3\) and \(b = 6i\) into the formula:\[ (-3)^2 - 2(-3)(6i) + (6i)^2 \]

Step 3: Calculate each term

1. Calculate \((-3)^2\): \[(-3)^2 = 9\]2. Calculate \(-2(-3)(6i)\): \[-2 \times (-3) \times 6i = 36i\]3. Calculate \((6i)^2\) (remembering \(i^2 = -1\)): \[(6i)^2 = 36i^2 = 36(-1) = -36\]

Step 4: Combine the calculated terms

Add the results from Step 3:\[9 + 36i - 36\]Simplify by combining like terms, starting with the real parts:\[9 - 36 = -27\]So, the expression becomes:\[-27 + 36i\]

Step 5: Express in standard form

The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The simplified expression from Step 4, \[-27 + 36i\]is already in standard form. Therefore, the complex number's real part is \(-27\) and the imaginary part is \(36i\).

Key Concepts

Binomial Expansion

The Binomial Expansion is a process used to expand expressions that are raised to a power. In this case, we are dealing with a binomial squared, such as \((-3 - 6i)^2\). To solve this, we utilize the binomial square formula:

• \((a - b)^2 = a^2 - 2ab + b^2\),

where \(a\) and \(b\) are parts of the binomial. It allows us to expand the square of the expression systematically.

Let's break it down:

• Identify \(a\) and \(b\): in this example, \(a = -3\) and \(b = 6i\).
• Substitute these values into the binomial formula.
• Calculate each of the terms separately.

Knowing how to expand binomials is a fundamental skill that simplifies working with polynomial equations.

Imaginary Numbers

Imaginary Numbers are a crucial component when dealing with complex numbers. They are based on the imaginary unit denoted by \(i\), defined as \(i^2 = -1\).

In the given problem, we encounter \(6i\), an imaginary number where 6 is the coefficient of \(i\).

• When squaring an imaginary number like \((6i)^2\), remember that \(i^2\) becomes \(-1\), making the result \(36(-1)=-36\).

Imaginary numbers play a significant role in complex equations, as they provide a means to include solutions that cannot be expressed as real numbers alone. Understanding imaginary numbers prepares you for more advanced topics in mathematics and engineering.

Standard Form of Complex Numbers

The Standard Form of Complex Numbers is expressed as \(a + bi\), where \(a\) is the real part and \(bi\) the imaginary part.

For the exercise, the expression simplifies to

• \(-27 + 36i\)

with

• \(-27\) as the real part,
• \(36i\) as the imaginary part.

This form is essential as it allows for easy identification and separation of the real and imaginary components of a complex number.

Expressing complex numbers in standard form is a critical skill since it facilitates analysis and application in various fields, such as electronics and mechanics, where complex numbers often represent waves and signals.