Problem 88 - Algebra A Combined Function

Question

Factor each expression completely. $$ x^{2}-\frac{1}{25} $$

Step-by-Step Solution

Answer

\((x + \frac{1}{5})(x - \frac{1}{5})\)

Step 1: Identify the form of the expression

The given expression is \( x^2 - \frac{1}{25} \). This expression is in the form of a difference of squares, specifically \( a^2 - b^2 \), where \( a = x \) and \( b = \frac{1}{5} \) because \( (\frac{1}{5})^2 = \frac{1}{25} \).

Step 2: Apply the difference of squares formula

The formula for the difference of squares is \( a^2 - b^2 = (a + b)(a - b) \). Using this formula, we substitute \( a = x \) and \( b = \frac{1}{5} \) into the equation.

Step 3: Factor the expression

Applying the difference of squares formula gives us: \[ x^2 - \frac{1}{25} = \left(x + \frac{1}{5}\right)\left(x - \frac{1}{5}\right) \]. This is the completely factored form of the expression.

Key Concepts

Difference of Squares

The concept of difference of squares is a vital part of algebra, especially when it comes to factoring. This technique is applied when an expression has the structure of \( a^2 - b^2 \). Here, "difference" means subtraction, and "squares" means that both numbers are squared.

• The standard form of the difference of squares is \( a^2 - b^2 \) where \( a \) and \( b \) are any algebraic terms.
• This expression can be factored as \( (a + b)(a - b) \).

In the given exercise, the expression \( x^2 - \frac{1}{25} \) is a perfect example of the difference of squares. The term \( x^2 \) represents \( a^2 \) and \( \frac{1}{25} \) corresponds to \( b^2 \). By recognizing this pattern, you can confidently apply the factoring technique.

Algebra

Algebra is a branch of mathematics dealing with symbols and rules for manipulating these symbols. It includes everything from solving simple equations to complex polynomial expressions.

• Algebraic expressions are composed of variables, coefficients, and constants.
• Factoring is a key algebraic technique to simplify expressions and solve equations.

When working with expressions like \( x^2 - \frac{1}{25} \), it's crucial to be comfortable with concepts such as variables (e.g., \( x \)) and constants (e.g., \( \frac{1}{25} \)). Understanding how to manipulate these components through operations like factoring can simplify many algebraic tasks, making it easier to handle more complicated problems later on. This exercise helps build these foundational skills effectively.

Factoring Techniques

Factoring is an essential algebraic technique used to break down or "factor" an expression into simpler terms or 'factors'. Various methods of factoring exist, each applicable to different types of expressions.

• The difference of squares technique is one of the simplest and most common methods, appropriate when the expression has the form \( a^2 - b^2 \).
• Other techniques include factoring by grouping and using the greatest common factor (GCF).

For the expression \( x^2 - \frac{1}{25} \), the difference of squares technique is used, turning the expression into \( \left(x + \frac{1}{5}\right)\left(x - \frac{1}{5}\right) \). Knowing when and how to apply these techniques can greatly enhance your problem-solving abilities in algebra. Mastery of these skills prepares students for more complex mathematics topics.