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Chapter 0: Problem 48

Add or subtract as indicated. $$\frac{x+3}{x-3}+\frac{x-3}{x+3}$$

Short Answer

Expert verified
The simplified expression of the equation is \(\frac {2x^2 + 18}{(x-3)(x+3)}\).

Step by step solution

01

Determine the Common Denominator

In this case, to add the two rational expressions, we need to find the common denominator. As the denominators are \(x-3\) and \(x+3\) respectively, their product \( (x-3)(x+3) \) will act as the common denominator.
02

Restructure Rational Numbers with Common Denominator

Rational expressions should be rewritten using the common denominator. Multiply each rational expression by an appropriate factor such that each fraction will have the same denominator, i.e., \(x-3\) should be multiplied to the second fraction and \(x+3\) to the first. And those will give \[(x+3) \cdot \frac{x+3}{x-3} + (x-3) \cdot \frac{x-3}{x+3}\] which simplifies to \[\frac{(x+3)^2}{(x-3)(x+3)} + \frac{(x-3)^2}{(x-3)(x+3)}\].
03

Combine Terms and Simplify

Next, add the numerators of these two fractions, remember the denominator still stays the same, so that results in \[\frac{(x+3)^2 + (x-3)^2}{(x-3)(x+3)}.\] This can be further expanded and simplified as \[\frac {x^2 + 6x + 9 + x^2 -6x +9}{(x-3)(x+3)} \Rightarrow \frac {2x^2 + 18}{(x-3)(x+3)}.\]

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