Problem 9 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$(5-2 i)^{2}$$

Step-by-Step Solution

Answer

The expression is \(21 - 20i\).

Step 1: Expand the Expression

Apply the formula for the square of a binomial. When you have \((x - y)^2 = x^2 - 2xy + y^2\), set \(x = 5\) and \(y = 2i\). Therefore, it becomes \((5 - 2i)^2 = 5^2 - 2(5)(2i) + (2i)^2\).

Step 2: Calculate Each Term

Calculate each part of the expansion separately:1. \(5^2 = 25\).2. \(-2 \times 5 \times 2i = -20i\).3. \((2i)^2 = 4i^2\). Since \(i^2 = -1\), this becomes \(4(-1) = -4\).

Step 3: Combine the Results

Put all the calculated terms together: \(25 - 20i - 4\). Combine the real parts (\(25\) and \(-4\)) to get \(21\). The imaginary part remains \(-20i\).

Step 4: Write in the Form \(a + b i\)

Combine and rewrite the result as \(21 - 20i\), identifying \(a = 21\) and \(b = -20\).

Key Concepts

Binomial Expansion

Binomial expansion is a process used to expand expressions that are raised to any given power. In the case of complex numbers, it involves the use of binomial theorem.
The binomial theorem provides a formula to expand expressions of the form \( (x + y)^n \),

but for the purpose of our simple squaring binomial, we use:

• \( (x - y)^2 = x^2 - 2xy + y^2 \)

In the original exercise, we expanded \( (5 - 2i)^2 \) using this formula. It helps break down complex expressions into simpler parts, which can easily be calculated. This process is invaluable when dealing with more complicated algebraic expressions.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a mathematical concept used to represent numbers that are not real. By definition, \( i \) is the square root of \( -1 \), which means \( i^2 = -1 \).This unusual property makes imaginary numbers unique. They are used to form complex numbers, which come in the shape of \( a + bi \).
In the given problem, we worked with \( 2i \), where you multiplied it by itself, resulting in \((2i)^2 = 4i^2\).

This then simplifies to \(-4\) since \( i^2 = -1 \). Understanding \( i \)'s properties is crucial when learning complex arithmetic.

Algebraic Expressions

Algebraic expressions involving complex numbers require careful handling. These expressions can have real and imaginary components. When expanding or simplifying, it is important to treat each part distinctly.For example, in \( 5^2 \),\quad \( -2 \times 5 \times 2i \),\quad \text{and}\quad (2i)^2,\ we tackled each term independently:

• Square the real part: \( 5^2 = 25 \)
• Multiply components: \( -20i \)
• Square the imaginary part with care due to \( i^2 = -1\), resulting in \(-4\)

After evaluating each term, we combine to get the final result. It's a systematic approach to simplifying and solving equations involving complex numbers.

Complex Conjugate

A complex conjugate is a pair that involves flipping the sign of a complex number's imaginary part. It is written by replacing \( a + bi \) with \( a - bi \).While we didn’t use conjugates directly in our existing problem, understanding them is key to many algebraic operations such as division or finding magnitude.

• Conjugates help simplify complex fractions
• They are used in rationalizing denominators
• They can find roots of complex numbers

Mastering the concept of the complex conjugate is fundamental for anyone diving deeper into complex number algebra.