Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
![### Simplify the Expression
Given the expression:
\[ \frac{2x}{x^2 + 4x - 12} - \frac{4}{x^2 + 4x - 12} \]
You are asked to simplify this expression.
**Steps to Simplify:**
1. **Identify the Common Denominator:**
The denominators \( x^2 + 4x - 12 \) for both terms are the same.
2. **Combine the Fractions:**
Since the denominators are the same, you can combine the numerators over the common denominator.
\[
\frac{2x - 4}{x^2 + 4x - 12}
\]
3. **Simplify the Numerator:**
Factor out the greatest common factor (GCF) in the numerator.
\[
2x - 4 = 2(x - 2)
\]
This gives:
\[
\frac{2(x - 2)}{x^2 + 4x - 12}
\]
4. **Factor the Denominator:**
Factor the quadratic expression in the denominator.
\[
x^2 + 4x - 12 = (x + 6)(x - 2)
\]
So, the expression becomes:
\[
\frac{2(x - 2)}{(x + 6)(x - 2)}
\]
5. **Cancel Common Factors:**
Notice that \( x - 2 \) is a common factor in both the numerator and the denominator.
\[
\frac{2 \overbrace{(x - 2)}^{\text{cancel this}}}{(x + 6)\overbrace{(x - 2)}^{\text{cancel this}}}
\]
Cancel \(x - 2\) from the numerator and the denominator.
\[
\frac{2}{x + 6}
\]
The simplified form of the given expression is:
\[ \frac{2}{x + 6} \]
Please note when simplifying, \( x \neq 2 \) and \( x \neq -6 \) to avoid division by zero.
Understanding these steps will help you simplify similar algebraic expressions efficiently.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2d158991-3671-4459-aef6-bd0615668acc%2F5c56848b-cbc2-4a54-8725-41c0085f730f%2Fcm766sm_processed.png&w=3840&q=75)
Transcribed Image Text:### Simplify the Expression
Given the expression:
\[ \frac{2x}{x^2 + 4x - 12} - \frac{4}{x^2 + 4x - 12} \]
You are asked to simplify this expression.
**Steps to Simplify:**
1. **Identify the Common Denominator:**
The denominators \( x^2 + 4x - 12 \) for both terms are the same.
2. **Combine the Fractions:**
Since the denominators are the same, you can combine the numerators over the common denominator.
\[
\frac{2x - 4}{x^2 + 4x - 12}
\]
3. **Simplify the Numerator:**
Factor out the greatest common factor (GCF) in the numerator.
\[
2x - 4 = 2(x - 2)
\]
This gives:
\[
\frac{2(x - 2)}{x^2 + 4x - 12}
\]
4. **Factor the Denominator:**
Factor the quadratic expression in the denominator.
\[
x^2 + 4x - 12 = (x + 6)(x - 2)
\]
So, the expression becomes:
\[
\frac{2(x - 2)}{(x + 6)(x - 2)}
\]
5. **Cancel Common Factors:**
Notice that \( x - 2 \) is a common factor in both the numerator and the denominator.
\[
\frac{2 \overbrace{(x - 2)}^{\text{cancel this}}}{(x + 6)\overbrace{(x - 2)}^{\text{cancel this}}}
\]
Cancel \(x - 2\) from the numerator and the denominator.
\[
\frac{2}{x + 6}
\]
The simplified form of the given expression is:
\[ \frac{2}{x + 6} \]
Please note when simplifying, \( x \neq 2 \) and \( x \neq -6 \) to avoid division by zero.
Understanding these steps will help you simplify similar algebraic expressions efficiently.
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