Problem 46 - Algebra A Combined Function

Question

Multiply. Write your answers in the form \(a+b i\). $$ (6-3 i)^{2} $$

Step-by-Step Solution

Answer

The expression \((6 - 3i)^2\) simplifies to \(27 - 36i\).

Step 1: Identify the Formula for Squaring a Binomial

To solve \((6 - 3i)^2\), we start by identifying the mathematical formula to use. The expression is in the form \((a - b)^2\), which equals \(a^2 - 2ab + b^2\). Here, \(a = 6\) and \(b = 3i\).

Step 2: Square the Real Component

Now, compute \(a^2\). For \(a = 6\), we have: \(6^2 = 36\).

Step 3: Multiply and Double the Real and Imaginary Components

Next, compute \(2ab\), where \(a = 6\) and \(b = 3i\). This gives us \(2 \times 6 \times 3i = 36i\).

Step 4: Square the Imaginary Component

Now, find \(b^2\), where \(b = 3i\). We calculate \((3i)^2 = 9i^2\). Recall that \(i^2 = -1\), so \(9i^2 = -9\).

Step 5: Combine All Parts Together

Combine the results from Steps 2, 3, and 4. We have: \(a^2 - 2ab + b^2 = 36 - 36i - 9\).

Step 6: Simplify the Expression

Simplify the combined expression: \(36 - 36i - 9 = 27 - 36i\). Thus, the expression in the form of \(a + bi\) is \(27 - 36i\).

Key Concepts

Binomial Expansion

When we talk about binomial expansion, we're exploring the process of expanding expressions that are raised to a power. A binomial expression is simply an algebraic expression that contains two terms. For example,

• In our exercise, we have \((6 - 3i)^{2}\),
• This expression is an example of a binomial as it combines two terms: a real number and an imaginary number.

To expand a binomial expression that’s squared, we employ the specific formula: \((a - b)^{2} = a^{2} - 2ab + b^{2}\). This formula helps in breaking down each component of the binomial so they can be easily squared and multiplied. Using this formula streamlines the process, allowing for systematic expansion and ensures accuracy. Remember, practicing this can greatly enhance your skills in algebra and make dealing with more complex expressions less daunting.

Imaginary Numbers

Imaginary numbers might sound a bit whimsical, but they're actually a crucial part of mathematics! In simple terms, imaginary numbers are numbers that give a negative product when squared. The unit for imaginary numbers is represented by \(i\), which is defined as: \(i^{2} = -1\). Here's a quick breakdown of what you need to know:

• Imaginary numbers appear when we take the square root of a negative number. It wasn’t possible until mathematicians introduced \(i\) as a solution.
• In our problem, the term \(3i\) is the imaginary component of our binomial expression.
• When handling operations involving imaginary numbers, remember that the imaginary unit obeys the rule \(i^{2} = -1\), which is often utilized to simplify expressions.

Imaginary numbers, along with real numbers, form the set of complex numbers, which further extends the number system and allows for solutions that were previously unreachable.

Algebraic Expressions

Algebraic expressions are the heart of algebra. They're constructed using numbers and variables connected by operations like addition, subtraction, multiplication, and division. In our scenario, \((6 - 3i)^{2}\) is an algebraic expression that includes both a real number and an imaginary number as components.

• Understanding algebraic expressions involves learning how to manipulate and simplify them based on mathematical rules.
• Common tasks include expanding, factoring, and simplifying.

To work effectively with algebraic expressions:

• Recognize the different parts: coefficients, variables, constants, and in cases like this, the imaginary unit.
• Apply the right formulas, as we did with applying the binomial expansion formula.
• Keep in mind the order of operations to ensure each term is manipulated correctly.

Handling algebraic expressions becomes simpler with practice, so keep solving different problems to master the art of managing them!