Problem 40 - Intermediate Algebra

Question

Multiply. Write the product in the form \(a+b i .\) See Example 4. $$ (6-3 i)^{2} $$

Step-by-Step Solution

Answer

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

Step 1: Recall the Formula for Squaring a Binomial

The expression given is \[(6 - 3i)^2\]Recall the formula for squaring a binomial: \[(a-b)^2 = a^2 - 2ab + b^2\].

Step 2: Identify a and b in the Expression

In the expression \[(6 - 3i)^2\],we identify \[a = 6\] and \[b = 3i\].

Step 3: Calculate the First Term in the Formula

Calculate \(a^2\): \[a^2 = 6^2 = 36\].

Step 4: Calculate the Second Term in the Formula

Calculate \(-2ab\): \[-2ab = -2 \times 6 \times 3i = -36i\].

Step 5: Calculate the Third Term in the Formula

Calculate \(b^2\): \[(3i)^2 = 9i^2 = 9(-1) = -9\]. Here, use the fact that \(i^2 = -1\).

Step 6: Combine All Terms

Combine the terms from Steps 3, 4, and 5: \[36 - 36i - 9\].

Step 7: Simplify the Expression

Simplify the expression: \[36 - 9 = 27\], resulting in \[27 - 36i\].

Key Concepts

Binomial Squaring

Squaring a binomial is a fundamental operation in algebra that includes using a specific formula. When you have an expression like

• a binomial of the form \((a - b)^2\), you can square it using the formula:
• \((a - b)^2 = a^2 - 2ab + b^2\).

This formula allows us to expand a squared binomial into a trinomial, making calculations more straightforward. Each term in the resulting expression corresponds to parts of the original binomial:

• The first term, \(a^2\), is the square of the first term in the binomial.
• The second term, \(-2ab\), involves multiplying both terms together, doubling the result, and maintaining the sign.
• The third term, \(b^2\), is the square of the second term in the binomial.

For example, in the complex expression \((6 - 3i)^2\), the values are identified as \(a = 6\) and \(b = 3i\). Squaring this binomial involves calculating each term using the formula to easily find the expanded result.

Imaginary Unit i

The imaginary unit \(i\) plays a crucial role in mathematics, especially in complex numbers. It is defined as \(i \equiv \sqrt{-1}\). This definition sets up the basis for complex arithmetic, most notably the fact that

• \(i^2 = -1\).

This means any power of \(i\) can be simplified by repeatedly applying the rule \(i^2 = -1\). For example, in the calculation of \((3i)^2\) in the original problem:

• The expression \((3i)^2\) is expanded into \(9i^2\).
• Since \(i^2=-1\), substituting gives \(9(-1) = -9\).

The presence of \(i\) in calculations allows us to square negative numbers, representing them as complex numbers, expanding the realm of mathematical solutions.

Simplifying Complex Expressions

Simplifying complex expressions involves reducing them to their standard form, which is typically \(a + bi\), where \(a\) and \(b\) are real numbers. This process involves

• combining like terms,
• using properties of real and imaginary numbers,
• and applying arithmetic operations.

In the given exercise, after expanding the binomial \((6 - 3i)^2\), we obtained an expression: \[36 - 36i - 9\]. The next step is to combine the real parts to simplify the expression:

• The real numbers \(36\) and \(-9\) combine to give \(36 - 9 = 27\).
• The term with \(i\) remains unchanged as \(- 36i\).

Thus, the simplified expression becomes \(27 - 36i\). By organizing complex expressions in this form, they can be easily interpreted and used in further calculations or solutions.