Problem 2 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$x^{2}-9$$

Step-by-Step Solution

Answer

The expression \(x^2 - 9\) factors to \((x - 3)(x + 3)\).

Step 1: Identify the Difference of Squares

The expression given is \(x^2 - 9\). Here, we have a square term \(x^2\) and \(9\), which is \(3^2\). Both of these terms are perfect squares, and the expression is in the form \(a^2 - b^2\).

Step 2: Apply the Difference of Squares Formula

The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). In our case, \(a\) is \(x\) and \(b\) is \(3\).

Step 3: Substitute into the Formula

Substitute \(a = x\) and \(b = 3\) into the formula to get \((x - 3)(x + 3)\).

Step 4: Verify the Factorization

Multiply \((x - 3)(x + 3)\) to ensure it equals the original expression. \((x - 3)(x + 3) = x^2 + 3x - 3x - 9 = x^2 - 9\), confirming the factorization is correct.

Key Concepts

Factoring

Factoring is a technique used in algebra to simplify expressions or solve equations. It involves expressing a polynomial as a product of its factors. One of the most important methods is the difference of squares, which is a particular type of factoring. This method applies to expressions that can be written in the form \(a^2 - b^2\). The difference of squares rule states that such an expression can be factored as:

• \(a^2 - b^2 = (a - b)(a + b)\)

In our original problem, the expression \(x^2 - 9\) fits the pattern of a difference of squares. Here, \(x^2\) is the square of \(x\) and \(9\) is the square of \(3\). By identifying these squares, you can then directly apply the factorization formula to break down the expression into two binomial factors: \((x - 3)\) and \((x + 3)\). This approach makes solving certain algebraic expressions much more manageable, and it's crucial to practice recognizing patterns like these.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They form the basic building blocks of algebra, allowing us to generalize and solve a variety of mathematical problems. Each part of an algebraic expression - like terms, coefficients, or variables - plays a specific role in defining the expression's behavior.A critical skill in working with algebraic expressions is manipulating them through operations like addition, subtraction, and factoring. In our case, factoring an algebraic expression such as \(x^2 - 9\) into its components

• \((x - 3)(x + 3)\)

can provide insights into the expression's structure and properties. Recognizing when expressions can be simplified through techniques like the difference of squares helps streamline solving equations and handling complex algebraic tasks. Always ensure that each step when manipulating algebraic expressions retains equality, checking the outcomes by reversing operations such as multiplying to verify factorizations.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically taking the form \(ax^2 + bx + c = 0\). These equations are prevalent in numerous areas of mathematics and applied sciences. Learning to solve quadratic equations involves several methods, such as factoring, completing the square, and the quadratic formula.In our example of factoring \(x^2 - 9\), although not set equal to zero, demonstrates the power of factoring in simplifying quadratic expressions. By expressing \(x^2 - 9\) as \((x - 3)(x + 3)\), we can solve equations assuming the expression is set to zero.

• \((x - 3)(x + 3) = 0\)

From here, you can find the solutions by setting each factor to zero, obtaining \(x = 3\) and \(x = -3\). Thus, understanding how to factor quadratics not only simplifies expressions but also helps solve quadratic equations efficiently, making it an invaluable skill in algebra.