Problem 35 - Beginning and Intermediate Algebra

Question

Factor completely. $$x^{2}-9$$

Step-by-Step Solution

Answer

The completely factored expression is \((x + 3)(x - 3)\).

Step 1: Identify the expression as difference of squares

The given expression can be written in the form \(a^2 - b^2\), where \(a = x\) and \(b = 3\). This is because \(x^2 - 9\) can be rewritten as \(x^2 - 3^2\).

Step 2: Apply the difference of squares formula

Using the difference of squares formula, we can factor the expression as follows: \(a^2 - b^2 = (a + b)(a - b)\) Substitute the values of a and b (a = x and b = 3) in the formula:

Step 3: Factor the expression

\(x^2 - 9 = (x + 3)(x - 3)\) So, the completely factored expression is \((x + 3)(x - 3)\).

Key Concepts

Difference of Squares

The difference of squares is a fundamental concept in algebra, which refers to an expression that can be represented in the form \(a^2 - b^2\). This formula is particularly useful because it can be factored into the product of two binomials. The general formula for factoring a difference of squares is:\[a^2 - b^2 = (a + b)(a - b).\]
When applying this concept, it is essential to correctly identify the terms \(a\) and \(b\). In the expression \(x^2 - 9\), \(a\) equals \(x\) and \(b\) equals 3, given that 9 is \(3^2\). Recognizing this pattern simplifies the process of factoring complex algebraic expressions, allowing each polynomial to be broken down into simpler components. By understanding difference of squares, students can gain a solid foundation for solving more advanced algebraic problems.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators (like addition and multiplication). They form the building blocks of algebra, enabling us to express mathematical ideas succinctly. An algebraic expression can consist of a single term (e.g., \(5x\)) or multiple terms (e.g., \(3x^2 - 4x + 7\)).
In the case of the expression \(x^2 - 9\), it is made up of two terms: \(x^2\) and \(-9\). Each term in an expression has its part to play—\(x^2\) represents a variable squared, and \(-9\) is a constant term. When manipulating algebraic expressions, it's important to apply rules and properties, such as distributive, associative, and commutative properties, to simplify or factor them. Recognizing patterns like the difference of squares can make these processes much more efficient.

Quadratic Expressions

Quadratic expressions are a specific type of polynomial that typically take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. These expressions are notable for their characteristic curve when graphed, known as a parabola.
A perfectly simple example of a quadratic expression is the one we are factoring, \(x^2 - 9\). In this particular case, it is already set with \(a = 1\), \(b = 0\), and \(c = -9\). The goal with such expressions often involves factoring them into more manageable binomial forms. Understanding how to factor quadratics is pivotal, as it is frequently used to solve equations, simplify expressions, and find x-intercepts of a graph. Mastery of quadratic expressions enables tackling a variety of algebraic challenges with confidence.