Problem 53 - College Algebra

Question

At the northwest corner of the park, the city is going to install a fountain. The area of the base of the fountain is \(9 x^{2}-25 \mathrm{~m}^{2}\). Factor the area to fi \(\mathrm{d}\) the lengths of the sides of the fountain.

Step-by-Step Solution

Answer

The sides are \(3x + 5\) and \(3x - 5\).

Step 1: Identify the Expression to Factor

Given the expression for the area of the base of the fountain is \(9x^2 - 25\). Our task is to factor this expression to find the dimensions of the fountain's base.

Step 2: Recognize a Difference of Squares

The expression \(9x^2 - 25\) is a difference of squares. A difference of squares takes the form \(a^2 - b^2\) and is factored as \((a + b)(a - b)\).

Step 3: Determine the Squares

We need to identify \(a^2\) and \(b^2\) such that the expression fits the difference of squares pattern.- \(9x^2\) can be written as \((3x)^2\).- \(25\) can be written as \(5^2\).

Step 4: Apply the Difference of Squares Formula

Substitute the values into the difference of squares formula:\[(3x)^2 - 5^2 = (3x + 5)(3x - 5)\]. This represents the factored form of the area expression.

Step 5: Interpret the Factored Form

The factored expression \((3x + 5)(3x - 5)\) indicates that the lengths of the sides of the fountain's base are \(3x + 5\) and \(3x - 5\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a way of expressing a polynomial as a product of its simpler factors, which, when multiplied together, give the original polynomial. It's an essential skill in algebra that involves recognizing patterns in expressions to simplify them.

A common method of factoring is by recognizing special algebraic identities, such as the difference of squares. In this process, you'll learn how to break down complex polynomials into simpler terms, which makes solving equations easier. In the exercise, the polynomial given is 9x^2 - 25, a quadratic expression that can be factored using rules for differences of squares.

Some strategies for factoring include:

• Identifying common factors in the terms.
• Recognizing and applying standard factorization formulas.
• Checking each step by multiplying the factors to ensure they give back the original polynomial.

With practice, factoring polynomials becomes an intuitive and straightforward tool for simplifying algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (such as addition or subtraction) that represent a value. In algebra, you'll frequently work with various forms of expressions that require simplification or factoring.

In the context of the difference of squares, understanding algebraic expressions involves recognizing patterns within the expression that facilitate easier manipulation. For example, the expression 9x^2 - 25 involves both numeric and variable components that fit into a standard algebraic identity, allowing for simple factorization.

Key aspects of working with algebraic expressions include:

• Identifying like terms that can be simplified.
• Utilizing operator rules to manipulate the expression.
• Recognizing forms such as a^2 - b^2, which imply specific factorization paths.

By mastering these elements, simplifying algebraic expressions becomes an essential skill, enhancing problem-solving efficiency in more complex algebraic problems.

Quadratic Expressions

Quadratic expressions are a type of polynomial that include a term squared, such as ax^2 + bx + c. These expressions are foundational in algebra and appear frequently in various mathematical problems, including area calculations and motion equations.

A classic example of a quadratic expression is when it's arranged in the form of a difference of squares, like in the exercise where we have 9x^2 - 25. The benefit here is such quadratic expressions can be quickly factored when recognized correctly.

Key points for dealing with quadratic expressions include:

• Understanding the general form: ax^2 + bx + c.
• Identifying special patterns such as a^2 - b^2 that simplify the factorization process.
• Re-writing quadratic expressions in factored form often reveals critical values like roots or zeroes.

Familiarity with these concepts allows you to handle quadratic equations and expressions effectively, paving the way for more complex problem-solving tasks in algebra.