Problem 28 - Algebra 2 and Trigonometry

Question

In \(26-37,\) find each product. $$ (2-3 i)^{2} $$

Step-by-Step Solution

Answer

The product is \(-5 - 12i\).

Step 1: Write the expression in expanded form

To find \((2-3i)^2\), start by writing it as a multiplication expression: \((2-3i)\times(2-3i)\). This is the base for the next steps.

Step 2: Use the distributive property (FOIL method)

Apply the distributive property (also known as the FOIL method) to \((2-3i)\times(2-3i)\): First multiply the first terms (\(2\times2\)), outer terms (\(2\times(-3i)\)), inner terms (\(-3i\times2\)), and last terms (\(-3i\times(-3i)\)).

Step 3: Calculate each part

Calculate each multiplication: \(2\times2 = 4\), \(2\times(-3i) = -6i\), \(-3i\times2 = -6i\), and \(-3i\times(-3i) = 9i^2\).

Step 4: Simplify using \(i^2 = -1\)

Remember that \(i^2 = -1\). This changes \(9i^2\) into \(-9\).

Step 5: Combine and simplify the expression

Combine all the parts: \(4 - 6i - 6i - 9\). This simplifies to \(4 - 9 - 12i\).

Step 6: Calculate final result

Continue the simplification to get \(-5 - 12i\). This is the product of \((2-3i)^2\).

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions involving multiplication over addition or subtraction. It's particularly useful in dealing with complex numbers. To apply this property, each term inside one set of parentheses is multiplied by each term inside another set of parentheses.
For example, in the expression \[ (2-3i) imes (2-3i) \] you use the distributive property to ensure each term in the first binomial is multiplied by each term in the second.

• First, multiply the first terms: \(2 \times 2 = 4\)
• Next, multiply the outer terms: \(2 \times (-3i) = -6i\)
• Then, the inner terms: \(-3i \times 2 = -6i\)
• Finally, the last terms: \(-3i \times (-3i) = 9i^2\)

This systematic process is sometimes referred to as the FOIL method, which stands for First, Outer, Inner, Last.

FOIL Method

The FOIL method is a mnemonic device used specifically for multiplying two binomials. The term FOIL stands for:

• **First**: Multiply the first terms in each binomial.
• **Outer**: Multiply the outermost terms in the product.
• **Inner**: Multiply the inside terms.
• **Last**: Multiply the last terms in each binomial.

In our example, \[ (2-3i) imes (2-3i) \] the FOIL method helps in systematically breaking down the multiplication process so nothing is overlooked. After using FOIL:

• First: \(2 \times 2 = 4\)
• Outer: \(2 \times (-3i) = -6i\)
• Inner: \(-3i \times 2 = -6i\)
• Last: \(-3i \times (-3i) = 9i^2\)

The FOIL method clarifies the multiplication steps by categorizing them into these four groups, making it easier to follow and simpler for learners to execute consistently.

Imaginary Unit i

The imaginary unit, often denoted as \(i\), is a unique concept fundamental to complex numbers. The defining characteristic of \(i\) is that it represents the square root of -1. This can often be counterintuitive because no real number squared will produce a negative result.
Key properties of the imaginary unit include:

• \(i = \sqrt{-1}\)
• \(i^2 = -1\)

Understanding these properties is crucial when working with complex numbers.
In our example of \[ (2-3i) \times (2-3i) \] the calculation \(-3i \times (-3i) = 9i^2\) requires applying the property \(i^2 = -1\) to convert \(9i^2\) into \(-9\). Understanding how to manipulate \(i\) and its powers simplifies handling complex numbers exponentially.

Simplification of Expressions

Simplifying expressions is the process of reducing them to their simplest form. This involves combining like terms, using known identities such as \(i^2 = -1\), and performing basic arithmetic operations.
In the example with \((2-3i)^2\), after applying the FOIL method:\[ 4 - 6i - 6i + 9i^2 \]you substitute \(9i^2\) with \(-9\), based on the property \(i^2 = -1\).
Then, you combine like terms:

• Real parts: \(4 - 9 = -5\)
• Imaginary parts: \(-6i - 6i = -12i\)

Thus, the simplified result is \(-5 - 12i\).
The key is to always look for opportunities to combine like terms and apply known identities. Simplification not only makes expressions easier to understand but also ready to be further analyzed or solved.