Problem 54 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. $$(6 k-3)^{2}$$

Step-by-Step Solution

Answer

The expanded form is \(36k^2 - 36k + 9\).

Step 1: Identify the Expression to Expand

The expression given is \[(6k - 3)^2\].This means we need to square the binomial \(6k - 3\).

Step 2: Apply the Binomial Square Formula

The formula for squaring a binomial is:\((a - b)^2 = a^2 - 2ab + b^2\).Identify the parts of the binomial: \(a = 6k\) and \(b = 3\).

Step 3: Square the First Term

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

Step 4: Compute Twice the Product of the Terms

Find \(-2ab\):\[-2(6k)(3) = -36k\].

Step 5: Square the Second Term

Calculate \(3^2\):\[3^2 = 9\].

Step 6: Combine All Terms Together

Combine the results from the previous steps:\[36k^2 - 36k + 9\].This is the expanded form of the expression.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can consist of numbers, variables, and operations. They represent values and form the foundation of algebra. An algebraic expression can be as simple as a single number or variable, or as complex as multiple terms combined with operators.

For example, in the given expression \(6k - 3\), we have:

• 6 is the coefficient of the variable \(k\).
• \(k\) is a variable, which means it can represent different values.
• -3 is a constant term.

The combination of these elements presents an expression that can be manipulated. Understanding each part is crucial to performing operations, such as expansion.

Exploring the Binomial Theorem

The binomial theorem is a powerful tool used to expand expressions that are raised to a power. It is particularly useful for binomials, which are algebraic expressions with two terms. In this exercise, we are dealing with \(6k - 3\)^2, which fits this description perfectly.

The formula used for squaring a binomial is:

• \((a + b)^2 = a^2 + 2ab + b^2\)
• \((a - b)^2 = a^2 - 2ab + b^2\)

Applying this to our expression, we identify \(a\) as 6k and \(b\) as 3. The formula helps us systematically expand the binomial by squaring each term individually and computing the cross-term expressed as \(2ab\).

This expansion makes complex expressions more manageable, revealing the underlying values.

Performing Mathematical Operations

Mathematical operations involve applying basic arithmetic processes such as addition, subtraction, multiplication, and division to solve problems. In the case of expanding \(6k - 3\)^2, these operations come into play as follows:

• Squaring terms: First, we square each individual term. \(a = 6k\) results in \((6k)^2 = 36k^2\), and \(b = 3\) gives \(3^2 = 9\).
• Multiplication and subtraction: We calculate twice the product of the terms using \(-2ab = -2(6k)(3) = -36k\).
• Combining terms: Finally, add all the calculated values to form the expanded expression: \(36k^2 - 36k + 9\).

Breaking down each operation helps simplify the process and ensures that we follow logical steps in manipulation. These are essential skills for tackling algebraic expressions and equations.