Problem 53 - College Algebra with Corequisite Support

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 find the lengths of the sides of the fountain.

Step-by-Step Solution

Answer

Lengths of sides: \((3x - 5)\) m and \((3x + 5)\) m.

Step 1: Identify the expression

The problem provides an area for the base of the fountain, given as \(9x^2 - 25\). We need to factor this expression to find the lengths of the sides.

Step 2: Recognize the form of the expression

Observe that \(9x^2 - 25\) is a difference of squares. It can be written in the form \(a^2 - b^2\) where \(a = 3x\) and \(b = 5\).

Step 3: Apply the difference of squares formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Substituting \(a = 3x\) and \(b = 5\) into the formula gives: \((3x - 5)(3x + 5)\).

Step 4: Write the factored form

The factored expression for the area of the fountain base is \((3x - 5)(3x + 5)\). This means the lengths of the sides of the fountain are \(3x - 5\) meters and \(3x + 5\) meters.

Key Concepts

Difference of Squares

Factoring polynomials often involves recognizing special patterns. One such pattern is the difference of squares. This concept comes into play when you have an expression of the form \(a^2 - b^2\). The term "difference" signifies subtraction, and "squares" point to terms that are squared, like \(x^2\) or \(25\).

When you encounter an expression like \(9x^2 - 25\), it can be broken down using the difference of squares formula. This formula is \(a^2 - b^2 = (a-b)(a+b)\). It tells us that the expression can be factored into two binomials, one with a subtraction and one with an addition.

Recognizing a difference of squares allows you to factor these expressions easily and is a key tool in simplifying algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In this problem, the expression \(9x^2 - 25\) is an algebraic expression. It consists of two parts:

• The term \(9x^2\) which shows a variable \(x\) squared and multiplied by 9
• The constant term, \(25\)

The process of factoring is used to simplify these expressions and solve for unknown values. By rewriting expressions, you can unearth factors that reveal more information about the mathematical situation being described.

In this context, algebraic expressions represent physical measurements, like the area of the fountain. This makes understanding how to manipulate them crucial for solving real-world problems, such as determining lengths or other dimensions.

Factored Form

The factored form of an algebraic expression is a way of writing it as a product of its factors. In our example, the expression \(9x^2 - 25\) was factored into \((3x - 5)(3x + 5)\). This transformation from an expression to its factored form simplifies solving equations and provides insight into the properties of the expression.

To achieve factored form:

• Identify special patterns or use techniques like the difference of squares
• Rewrite the expression as a multiplication of simpler expressions (binomials)

Factorization is especially helpful when estimating dimensions or solving problems in geometry, such as finding the sides of a square or a rectangle. In this exercise, having the area factored allows us to identify possible lengths of the fountain's sides, illustrating the practical significance of this algebraic technique.