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Chapter 12: Problem 98

For what values of the variable is the rational expression undefined? $$\frac{x+2}{x^{2}-4}$$

Short Answer

Expert verified
The rational expression \(\frac{x+2}{x^{2}-4}\) is undefined for \(x = 2, -2\). These values make the denominator equal to zero, which is forbidden in mathematics because division by zero is undefined.

Step by step solution

01

Identify the denominator

The denominator of the given rational expression \(\frac{x+2}{x^{2}-4}\) is \(x^{2}-4\). This will be the focus since the rational expression is undefined wherever the denominator equals zero.
02

Solve the equation \(x^{2}-4 = 0\)

To find the values of \(x\) that make the denominator zero, set the denominator equal to zero. Now the equation will look like this: \(x^{2}-4 = 0\). Solve this equation for \(x\).
03

Applying difference of squares

In order to solve the equation, you recognize that the denominator can be factored as the difference of two squares (because \(4 = 2^{2}\)). So the equation \(x^{2}-4 = 0\) can be factored as \((x-2)(x+2) = 0\)
04

Solve for \(x\)

By setting each factor equal to zero and solving for \(x\), you find \(x = 2\) for \((x-2) = 0\) and \(x = -2\) for \((x+2) = 0\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Undefined Expressions
A rational expression is undefined wherever its denominator is zero. This is because division by zero is not allowed in mathematics; it does not yield a meaningful result. Therefore, when analyzing rational expressions, the first step is to identify the values of the variable that make the denominator equal to zero. This can be done by setting the denominator equal to zero and solving for the variable. For example, if the denominator of the expression is \( x^2 - 4 \), you would solve the equation \( x^2 - 4 = 0 \) to find the values of \( x \) that make the expression undefined.
Difference of Squares
The difference of squares is a special factorization technique used in algebra when an expression is in the form \( a^2 - b^2 \). This can be rewritten as \((a - b)(a + b)\). For example, \( x^2 - 4 \) is a difference of squares because it can be written as \((x^2 - 2^2)\). This can then be factored into \((x - 2)(x + 2)\). Recognizing and applying difference of squares allows you to simplify expressions or solve equations more effectively, as it breaks the expression into simpler binomials.
Factoring
Factoring is the process of rewriting an expression as the product of its simpler components or factors. In the context of rational expressions, factoring is often used to simplify expressions or find the points where the denominator is zero. For instance, with the expression \( x^2 - 4 \), recognizing it as a difference of squares, you can factor it into \((x - 2)(x + 2)\). Once factored, it鈥檚 easier to identify the solutions to the equation, which in this case are the values of \( x \) that make the denominator zero: \( x = 2 \) and \( x = -2 \). Factoring is a key skill in algebra that can simplify solving equations and understanding expressions.

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Most popular questions from this chapter

Use the following information. Blotting paper is a thick, soft paper used for absorbing fluids such as water or ink. The distance \(d\) (in centimeters) that tap water is absorbed up a strip of blotting paper at a temperature of \(28.4^{\circ} \mathrm{C}\) is given by the equation \(d=0.444 \sqrt{t}\) where \(t\) is the time (in seconds). Approximately how many minutes would it take for the water to travel a distance of 28 centimeters up the strip of blotting paper?

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The vertices of a right triangle are \((0,0),(0,6),\) and (6, 0). What is the length of the hypotenuse? (A) 6 B) \(6 \sqrt{2}\) (C) 36 (D) 72

Use the following information. Blotting paper is a thick, soft paper used for absorbing fluids such as water or ink. The distance \(d\) (in centimeters) that tap water is absorbed up a strip of blotting paper at a temperature of \(28.4^{\circ} \mathrm{C}\) is given by the equation \(d=0.444 \sqrt{t}\) where \(t\) is the time (in seconds). How far up the blotting paper would the water be after \(33 \frac{1}{3}\) seconds?
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