Problem 31 - College Algebra

Question

In Exercises \(31-40,\) factor the difference of two squares. $$x^{2}-100$$

Step-by-Step Solution

Answer

The factored form of \(x^{2} - 100\) is \((x-10)(x+10)\).

Step 1: Identify the Difference of Squares

Identify that \(x^{2} - 100\) is a difference of squares. It can be rewritten as \(x^{2} - 10^{2}\). Here, \(a\) is \(x\) and \(b\) is \(10\).

Step 2: Apply the Formula

Apply the difference of squares formula, \(a^{2} - b^{2} = (a-b)(a+b)\). In this case, replace \(a\) with \(x\) and \(b\) with \(10\).

Step 3: Write the Simple Factored Form

The factorization of the expression \(x^{2} - 100\) becomes \((x-10)(x+10)\). This is the factored form of the given expression.

Key Concepts

Difference of Squares

The difference of squares is a special algebraic expression where two perfect squares are subtracted. For example, in the expression \(x^2 - 100\), both \(x^2\) and \(100\) are perfect squares. The number \(x^2\) is the square of \(x\), and \(100\) is the square of \(10\), written as \(10^2\).

This type of expression can always be factored using a specific rule called the "difference of squares formula." This formula states: \[a^2 - b^2 = (a-b)(a+b)\] where \(a\) and \(b\) are any real numbers.

Understanding this pattern helps simplify expressions and solve equations that involve subtraction of squares. By recognizing both terms as perfect squares, you can transform a complex expression into a simpler, factored form.

Algebra

Algebra involves working with symbols and rules for manipulating those symbols to solve equations. It allows you to express mathematical relationships in a generalized form. When dealing with the difference of squares, algebraic manipulation is used to simplify expressions.

This begins with recognizing patterns such as perfect squares and applying the correct formula. For example, in algebra, we start by identifying terms that are perfect squares, like \(x^2\) and \(100\) in \(x^2 - 100\).
This pattern recognition reduces complexity.

The process involves:

• Identifying terms as squares, like \(x^2\) and \(10^2\)
• Setting up the formula \(a^2 - b^2 = (a-b)(a+b)\)
• Substituting specific terms into the formula

This method helps in finding solutions quickly and efficiently, showcasing the power of algebra.

Factored Form

The factored form of a quadratic expression breaks down the equation into the product of its binomials. For the expression \(x^2 - 100\), the factored form is \((x-10)(x+10)\). This transformation involves using the difference of squares formula.

Factoring is crucial as it simplifies complex expressions and equations. It allows you to solve equations for their roots or zeros effectively.
This is particularly useful in graphing quadratic equations.

Key aspects of factored form include:

• The expression is split into two binomials, making it easier to solve
• The solutions to \((x-10)=0\) and \((x+10)=0\) give the roots \(x=10\) and \(x=-10\)
• Factored form is often the "simplest" form of an expression

Thus, understanding and applying the factored form enables efficient problem-solving in algebra.