Problem 126 - Algebra and Trigonometry

Question

Explain how to factor the difference of two squares. Provide an example with your explanation.

Step-by-Step Solution

Answer

Factoring the difference of squares involves applying the algebraic identity \(a^2 - b^2 = (a+b)(a-b)\). For example, \(9x^2 - 4\) factors to \((3x+2)(3x-2)\).

Step 1: Understanding the Concept

The difference of two squares is a special algebraic identity, which states that for any two numbers \(a\) and \(b\), the difference of their squares \(a^2 - b^2\) can be factored to \((a+b)(a-b)\). This identity helps simplify complex mathematical expressions or solve quadratic equations.

Step 2: Factoring Process

In order to factorize an expression of the form \(a^2 - b^2\), one identifies the terms which are perfect squares and applies the formula. Using the rule, \(a^2 - b^2\) is factored into \((a+b)(a-b)\). It's necessary to remember that the order of the terms in the factors does not matter, i.e., \((a+b)(a-b)\) is the same as \((a-b)(a+b)\).

Step 3: Example Explanation

For instance, consider the expression \(9x^2 - 4\). Here \(a = 3x\) and \(b = 2\) are each squares, so applying our formula, factored form of \(9x^2 - 4\) is \((3x+2)(3x-2)\).

Key Concepts

Factoring Algebraic Expressions

Understanding how to factor algebraic expressions is fundamental in algebra. Factoring simplifies expressions and solves equations. To factor an expression means to break it down into a product of simpler factors. For example, the expression \(8x^2 + 4x\) can be factored by extracting the common factor, which is \(4x\), resulting in \(4x(2x + 1)\).

When factoring expressions, look for common factors, special patterns like the difference of two squares or perfect squares, and use methods such as grouping or the box method. A factorized expression is easier to work with, especially when solving equations or simplifying complex fractions.

Algebraic Identities

Algebraic identities are equations that are true for all values of the variables involved. One of the most frequently used identities is the difference of two squares, which states that \(a^2 - b^2 = (a+b)(a-b)\). These identities serve as shortcuts to simplify expressions and solve equations more efficiently.

Other common algebraic identities include the square of a binomial \( (a+b)^2 = a^2 + 2ab + b^2 \) and the cube of a binomial \( (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \). Understanding these identities will greatly enhance the ability to rearrange and solve algebraic equations, making them an essential tool for students to master.

Solving Quadratic Equations

Quadratic equations, which are in the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, can be solved by various methods, including factoring, completing the square, or using the quadratic formula. When the equation can be factored into the difference of two squares, it simplifies the solving process.

With the factored form, set each factor equal to zero and solve for \(x\) to find the solutions of the equation. For the equation \(9x^2 - 4 = 0\), factoring gives us \(3x + 2)(3x - 2) = 0\), leading to the solutions \(x = -\frac{2}{3}\) and \(x=\frac{2}{3}\). Recognizing when a quadratic equation can be simplified using algebraic identities like the difference of two squares is a powerful technique in solving these types of problems efficiently.