Problem 60 - Elementary and Intermediate Algebra

Question

Factor.$$20 x^{2}-5$$

Step-by-Step Solution

Answer

The expression factors to \( 5(2x - 1)(2x + 1) \).

Step 1: Identify a Common Factor

First, we need to check if there is a common factor in the expression. The expression is \( 20x^2 - 5 \). Both terms, \( 20x^2 \) and \( 5 \), have a common factor of 5. Let's factor 5 out of the expression.

Step 2: Factor Out the Common Factor

Take out the 5 from each term in the expression: \( 20x^2 - 5 = 5(4x^2 - 1) \). Now the expression is simplified to \( 5(4x^2 - 1) \).

Step 3: Recognize the Difference of Squares

The expression inside the parenthesis \( 4x^2 - 1 \) is a difference of squares. Recall that a difference of squares can be factored as \( a^2 - b^2 = (a - b)(a + b) \). In this case, \( 4x^2 = (2x)^2 \) and \( 1 = 1^2 \).

Step 4: Apply the Difference of Squares Formula

Using the difference of squares formula, factor \( 4x^2 - 1 \) as \( (2x - 1)(2x + 1) \).

Step 5: Write the Complete Factorization

Combine the factored expression from steps 2 and 4: \( 5(4x^2 - 1) = 5(2x - 1)(2x + 1) \). This is the fully factored form of the original expression.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and at least one arithmetic operation. In our exercise above, you can see an expression like this: \(20x^2 - 5\).
These expressions are a core part of algebra and serve as the basis for creating equations and inequalities. Let's break them down further:

• Variables: These are symbols, usually letters, such as \(x\), which stand for unknown values or a range of values.
• Constants: Numbers like 5 in our expression, these have fixed values.
• Coefficients: Numbers like 20 are multiplied by the variable parts.

By understanding how these components interact, we can simplify expressions using various techniques such as factoring. This process often involves identifying patterns or manipulating terms to create a more workable form.

Exploring the Difference of Squares

The difference of squares is a specific pattern that appears frequently in algebra. It takes the form \(a^2 - b^2\) and can be easily factored into \((a - b)(a + b)\). Recognizing this pattern allows for a quick simplification of expressions.
In our example, inside the parentheses after step 2, we find \(4x^2 - 1\). This expression can be seen as \((2x)^2 - 1^2\).

• Calculating squares: Here, \(4x^2\) is the square of \(2x\), and 1 is the square of 1.
• Applying the formula: By using \(a = 2x\) and \(b = 1\), we can transfer \(a^2 - b^2\) into \((a - b)(a + b)\).

This technique is incredibly useful in algebra as it provides a straightforward method to factor and simplify otherwise complex expressions.

Identifying the Common Factor

Factoring by identifying a common factor is one of the first steps in simplifying algebraic expressions. It involves finding a number or variable present in each term of the expression.
In our example, \(20x^2 - 5\), the common factor is 5. Factoring this out simplifies the expression significantly.

• Step-by-step process: The terms \(20x^2\) and 5 both share the factor 5. Therefore, the expression can be rewritten as \(5(4x^2 - 1)\).
• Simplification: By factoring out the common factor, we transform the expression into a simpler form that can be further worked on, as we see with the difference of squares.

Understanding this process is key as it lays a foundational step in more advanced algebraic manipulations.