Problem 43 - Beginning and Intermediate Algebra

Question

Factor completely. $$c^{2}-\frac{9}{16}$$

Step-by-Step Solution

Answer

The short answer based on the given solution is: The completely factored form of the given expression \(c^2 - \frac{9}{16}\) is \((c+\frac{3}{4})(c-\frac{3}{4})\).

Step 1: Identify the expression as the difference of squares

The given expression is: \[c^2 - \frac{9}{16}\] By observing the given expression, we can identify that it is a difference of two squares. This is because: \[c^2 = (c)^2\] \[\frac{9}{16} = (\frac{3}{4})^2\]

Step 2: Apply the difference of squares formula

The difference of two squares formula states that for any two real numbers a and b: \[a^2 - b^2 = (a+b)(a-b)\] In our case, \(a=c\) and \(b=\frac{3}{4}\), so applying the difference of squares formula, we get: \[(c^2 - (\frac{3}{4})^2) = (c+\frac{3}{4})(c-\frac{3}{4})\]

Step 3: Simplify the expression

Since the expression cannot be simplified further, our final factored form is: \[(c+\frac{3}{4})(c-\frac{3}{4})\]

Key Concepts

Difference of Squares

The expression given in the exercise, \(c^2 - \frac{9}{16}\), is a classic example of the difference of squares. The difference of squares is a specific algebraic pattern utilized frequently in mathematics. It applies when you subtract one square number from another. This special pattern is represented by the formula:

• \(a^2 - b^2 = (a+b)(a-b)\)

The key to solving these expressions lies in recognizing when an expression fits this pattern.
In our case, the expression \(c^2\) is already a squared term and \(\frac{9}{16}\) can be rewritten as \((\frac{3}{4})^2\). Recognizing these perfect squares allows us to factor the expression quickly and accurately.

Algebraic Expressions

Algebraic expressions are fundamental in mathematics. They consist of numbers, variables, and operations (such as addition and subtraction). In our exercise, we deal with an algebraic expression consisting of a variable \(c\) and a constant term. This expression is:

• \(c^2 - \frac{9}{16}\)

This expression has two terms: one involving the variable \(c\) and another constant term.
To understand and manipulate these expressions, we apply rules and techniques like factoring. This allows us to simplify and solve equations or to transform them into a product of simpler expressions.
Algebraic expressions can represent a variety of real-world situations in mathematical form. Simplifying them, as we did with factoring, reveals deeper insights into their structure.

Factoring Techniques

Factoring techniques are essential tools in algebra for simplifying expressions. The goal of factoring is to express a polynomial as a product of its factors.
One common technique is factoring expressions that are a difference of squares, as seen in our exercise. Here, the formula \(a^2 - b^2 = (a+b)(a-b)\) is directly applied once the expression is identified as a difference of squares.

• Step 1: Identify square terms in the expression.
• Step 2: Apply the difference of squares formula by setting \(a\) and \(b\) to the square root of their respective expressions.

Other factoring techniques include grouping, factoring out the greatest common factor, and using special binomial products like the sum and difference of cubes.
Each technique has specific scenarios where it is best applied, helping to simplify, solve, and understand algebraic expressions more effectively.