Problem 82 - College Algebra with Modeling and Visualization

Question

Multiply the expressions. $$(9 x-4)(9 x+4)$$

Step-by-Step Solution

Answer

The product is \(81x^2 - 16\).

Step 1: Identify the Structure

The expression given is \((9x - 4)(9x + 4)\). Notice that it is in the form \((a - b)(a + b)\), which is a difference of squares pattern.

Step 2: Apply the Formula

The formula for a difference of squares is \((a-b)(a+b) = a^2 - b^2\). In this case, \(a = 9x\) and \(b = 4\).

Step 3: Compute \(a^2\)

Calculate \(a^2\) using \(a = 9x\). Thus, \(a^2 = (9x)^2 = 81x^2\).

Step 4: Compute \(b^2\)

Calculate \(b^2\) where \(b = 4\). Thus, \(b^2 = 4^2 = 16\).

Step 5: Substitute and Simplify

Substitute the computed values into the difference of squares formula: \(81x^2 - 16\). The simplified expression is \(81x^2 - 16\).

Key Concepts

Multiplying Expressions

When you multiply expressions, you are essentially distributing each term in one expression across every term in the other expression. This is a fundamental skill in algebra that helps you simplify equations and solve various types of problems.

To multiply expressions like (9x - 4) and (9x + 4), you use the distributive property, which involves multiplying each term in the first expression by each term in the second. This can be a bit more complex if both expressions have multiple terms, but there are strategies to simplify this work, such as identifying patterns like the difference of squares.

• Always look for recognizable patterns when multiplying expressions.
• Use distribution carefully by ensuring each term in the first expression multiplies with each term in the second.
• Check if special formulas or shortcuts, like the difference of squares, can simplify your work.

Difference of Squares Formula

The difference of squares formula is a powerful tool in algebra. It applies to expressions that are structured in a specific way: (a - b)(a + b). The formula states that (a - b)(a + b) = a^2 - b^2. This pattern is special because it allows you to quickly multiply the terms without going through every single distribution step.

In our example, the expressions (9x - 4) and (9x + 4) fit this pattern perfectly.

• Identify a = 9x and b = 4.
• Apply the formula by calculating a^2 = (9x)^2 which simplifies to 81x^2.
• Calculate b^2 = 4^2, resulting in 16.

Substitute these values into the formula to get 81x^2 - 16. The expression has now been simplified using the difference of squares.

Quadratic Expressions

Quadratic expressions are a type of polynomial that have the form ax^2 + bx + c. In this formula, "a", "b", and "c" represent constants, with "a" being non-zero. Quadratics are important because they form the basis of many algebraic concepts and applications.

While the result from multiplying a difference of squares, 81x^2 - 16, is not a typical quadratic because it lacks a linear 'bx' term, it is essential to recognize it as a "quadratic-like" expression due to its highest degree of two.

• Quadratics can be solved or simplified using various methods such as factoring, completing the square, or using the quadratic formula.
• A strong understanding of how to handle quadratic expressions supports advanced algebraic problem-solving.
• Always check the form and structure of an expression to apply the correct simplification or solving method.