Problem 78 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (5-3 i)^{2} $$

Step-by-Step Solution

Answer

\((5-3i)^2 = 16 - 30i\).

Step 1: Identify the expression

The given expression is \((5 - 3i)^2\). Our task is to expand this expression using algebra and express it in the standard form of a complex number, which is \(a + bi\).

Step 2: Apply the binomial expansion formula

Recall that \[(a - b)^2 = a^2 - 2ab + b^2\]Here, \(a = 5\) and \(b = 3i\). Substitute these values into the formula:\[(5 - 3i)^2 = 5^2 - 2 \cdot 5 \cdot 3i + (3i)^2\]

Step 3: Calculate each term

Calculate each part of the formula separately:- \(5^2 = 25\)- \(-2 \cdot 5 \cdot 3i = -30i\)- \((3i)^2 = (3^2)(i^2) = 9(-1) = -9\) because \(i^2 = -1\).

Step 4: Combine the terms

Now, add the calculated values together:\[25 - 30i - 9\]

Step 5: Simplify the expression

Simplify the addition:\[25 - 9 = 16\]So, the expression becomes \[16 - 30i\].

Step 6: Final Step: Express in standard form

The standard form of a complex number is \(a + bi\). Our final expression is already in this form:\[16 - 30i\].

Key Concepts

Binomial Expansion

Binomial expansion is a powerful tool in algebra that helps us expand expressions raised to a power. When we have a binomial expression like \((a - b)^2\), we use the binomial expansion formula: \[a^2 - 2ab + b^2\]. This formula is particularly useful to simplify and compute expressions involving complex numbers without multiplying manually.
To apply the binomial theorem for \((5 - 3i)^2\), we identify \(a = 5\) and \(b = 3i\). By replacing these values into the formula, we systematically expand the expression into manageable parts.

• Calculate \(a^2\) as \(5^2\), which gives \(25\).
• Calculate \(-2ab\) as \(-2 \cdot 5 \cdot 3i\), resulting in \(-30i\).
• Calculate \(b^2\) as \((3i)^2 = 9(-1) = -9\).

Combining these results: \[25 - 30i - 9\] becomes \[16 - 30i\]. This makes "Binomial Expansion" a structured approach for handling complex expressions.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a fundamental component of complex numbers. It is defined as the square root of \(-1\), which is not possible with real numbers.
The property \(i^2 = -1\) is pivotal to calculations involving complex numbers. In our exercise, we see this whenever we square \(3i\) to get \(9(-1) = -9\).
The imaginary unit allows us to work with equations that have no real solutions otherwise, for example, the equation \(x^2 + 1 = 0\). The introduction of \(i\) provides a broader number system called complex numbers, typically expressed as \(a + bi\).

• Real part: \(a\)
• Imaginary part: \(b\)

By understanding \(i\), one can explore many areas of math and physics where real numbers alone are insufficient.

Algebraic Expressions

Algebraic expressions involve numbers, variables, and operations such as addition, subtraction, multiplication, and division. When dealing with complex numbers, these expressions can appear intimidating at first, but are approachable with step-by-step solutions.
In our example, \((5 - 3i)^2\) is an algebraic expression involving a complex number. When handling such expressions:

• Recognize the structure of the expression.
• Apply appropriate algebraic principles or formulas, such as binomial expansion.
• Separate real and imaginary components.

By breaking down the expression into smaller parts, we simplify the process. For instance, after using binomial expansion, we ended up with \[25 - 30i -9\], simplifying further to \[16 - 30i\]. This illustrates how algebraic expressions help manage complex operations and lead us to solutions efficiently.