Problem 61 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((3 x-4)(3 x+4)\)

Step-by-Step Solution

Answer

The simplified expression is \(9x^2 - 16\).

Step 1: Identify the Formula to Use

The given algebraic expressions \((3x - 4)(3x + 4)\) suggest the use of the difference of squares formula, which states \((a-b)(a+b) = a^2 - b^2\).

Step 2: Apply the Difference of Squares Formula

Here, let \(a = 3x\) and \(b = 4\). Plug these values into the formula to get \((3x)^2 - 4^2\).

Step 3: Calculate Each Square in the Formula

First, calculate \((3x)^2\), which is \((3x)(3x) = 9x^2\). Next, calculate \(4^2\), which is \(16\).

Step 4: Simplify the Expression

Substitute the calculated values back into the formula: \(9x^2 - 16\), which is already simplified.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations that describe a mathematical idea or problem. In the exercise, we have two algebraic expressions in the form \(3x - 4\) and \(3x + 4\). These expressions can be thought of as instructions to calculate a value given a specific \(x\).
Understanding the structure of these expressions:

• Coefficient: In \(3x\), the number 3 is a coefficient, showing how many units of \(x\) there are.
• Variable: \(x\) is the variable that can change or be given a specific value.
• Constant: The numbers -4 and +4 are constants, values that don't change.

Working with algebraic expressions means cleverly interacting with these parts to find solutions or simplify them for clearer understanding.

Special Product Formula

Special product formulas in algebra are pre-established rules that help solve certain types of multiplication problems quickly. One specific formula we often use is the difference of squares: \(a^2 - b^2 = (a - b)(a + b)\).
This formula is especially useful because:

• It turns a multiplication of two binomials into a simple subtraction problem.
• By recognizing the pattern, complex calculations can be avoided.

In our example, we applied this to \(3x - 4\) and \(3x + 4\).
You identify the terms: \(a = 3x\) and \(b = 4\), then apply the formula to get \[a^2 - b^2 = (3x)^2 - 4^2\]. This matches the formula perfectly, making calculations almost immediate without needing each term to be multiplied individually.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to understand or use. This often means performing arithmetic with known numbers and combining like terms.
In the exercise, after recognizing and using the difference of squares formula, we end up with \(9x^2 - 16\).
Steps to simplify this include:

• Calculating individual squares: \(3x \, squared gives \, 9x^2\) and \(4^2 = 16\).
• Substituting these into the formula: We thus have \(9x^2 - 16\).

Here, the expression \(9x^2 - 16\) is already as simple as it can be because there are no like terms to combine or further arithmetic to perform. Simplified expressions are compact, making them easy to use in further calculations or comparisons.