Simplify: \frac{x - 2}{x^{2} + 4x - 12} A. \frac{1}{x + 6}; x not equal to -6 B. \frac{1}{x + 6};...

Question:

Simplify: {eq}\quad \displaystyle\frac{x - 2}{x^{2} + 4x - 12} {/eq}

A. {eq}\;\displaystyle\frac{1}{x + 6}; \quad x \neq -6 {/eq}

B. {eq}\;\displaystyle\frac{1}{x + 6}; \quad x \neq -6,2 {/eq}

C. {eq}\;\displaystyle\frac{1}{x + 2}; \quad x \neq -2 {/eq}

D. {eq}\;\displaystyle x + 2 {/eq}

Simplifying

Simplifying fractions that have quadratic terms in either the denominator or numerator or both is all about factoring the quadratic term(s). Once this is done, the common terms of the numerator and denominator can be removed.

Answer and Explanation:

The expression is simplified as follows.


$$\begin{align} &\frac{x - 2}{x^{2} + 4x - 12}\\ =&\frac{x-2}{x^2+6x-2x-12}\\ =&\frac{x-2}{x(x+6)-2(x+6)}\\ =&\frac{(x-2)}{(x-2)(x+6)}\\ =&\frac{1}{x+6} \end{align} $$


In the simplified expression, {eq}x\neq 6 {/eq} as when it takes the value of 6, the denominator becomes equal to 0 and the fraction becomes undefined. So, the correct answer is A.


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Practice Simplifying Algebraic Expressions

from ELM: CSU Math Study Guide

Chapter 6 / Lesson 7
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