Problem 52 - Beginning and Intermediate Algebra

Question

Factor completely. $$36 x^{2}-25$$

Step-by-Step Solution

Answer

The factored expression of \(36x^2 - 25\) is \((6x + 5)(6x - 5)\).

Step 1: Identify the difference of squares

First, we identify the difference of squares in the given expression: \(36x^2 - 25\) We can rewrite this as: \((6x)^2 - (5)^2\) Now we have our values for \(a\) and \(b\): \(a = 6x\) \(b = 5\)

Step 2: Apply the difference of squares rule

Now we apply the difference of squares rule to our expression: \(a^2 - b^2 = (a + b)(a - b)\) Using our values for \(a\) and \(b\): \((6x + 5)(6x - 5)\) So, the factored expression is: \((6x + 5)(6x - 5)\)

Key Concepts

Difference of Squares

The difference of squares is a special case in algebra where an expression is in the form of two perfect squares separated by a subtraction sign. Recognizing this pattern can simplify factoring.

The general form is: \[ a^2 - b^2 \]This expression can always be factored into:

• \((a + b)(a - b)\)

The key thing to remember is that both terms must be perfect squares. Perfect squares are numbers or expressions like 4, 9, 16, 25, or \(x^2\), \((3x)^2\), etc., which can be written as something raised to the power of 2.

In the exercise, the expression \(36x^2 - 25\) is a difference of squares because it can be rewritten as \((6x)^2 - (5)^2\). By identifying \(a = 6x\) and \(b = 5\), you can easily factor the expression following the difference of squares rule to get \((6x + 5)(6x - 5)\). This method is a powerful tool for quickly simplifying and solving polynomial expressions.

Algebra

Algebra is a fundamental part of mathematics dealing with symbols and the rules for manipulating those symbols. It is all about finding the unknown or putting real life variables into equations and then solving them.

When we talk about factoring in algebra, we are referring to expressing an equation or expression as a product of its factors, or simpler expressions. For example, factoring \(36x^2 - 25\) means breaking it down into multiplication of simpler factors: \((6x + 5)(6x - 5)\).

Some of the key operations in algebra, like addition, subtraction, multiplication, and division, are used to manipulate expressions and to isolate variables or solve for them. Mastery of these operations is crucial, as they form the backbone of solving many algebraic equations. In polynomial factoring, recognizing patterns such as the difference of squares is instrumental in simplifying complex expressions.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree 2. They generally take the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are coefficients. However, in some special cases, a quadratic expression may appear simply as the difference of two squares like in the exercise provided.

When we have expressions like \(x^2 - 16\), it is crucial to recognize it as a quadratic expression and often, expressions like these can be factored using straightforward methods like the difference of squares, resulting in \((x + 4)(x - 4)\). This not only simplifies the equation but can also help in solving for \(x\) when set equal to zero.

In the given exercise \(36x^2 - 25\), despite lacking a middle term, we can see it is a quadratic because \(6x\) raised to power two equates to the first term. Understanding the structure of quadratics and their associated factoring techniques like difference of squares equips you with essential skills for solving a broad array of mathematical problems efficiently.