QQuestionMathematics
QuestionMathematics
What is the difference?
\begin{aligned}
& \frac{x+ 5}{x+ 2}-\frac{x+ 1}{x^{2}+ 2 x} \\
& \text { (1) } \frac{x^{2}+ 4 x- 1}{x(x+ 2)} \\
& \text { (2) } \frac{x^{2}+ 4 x+ 1}{x(x+ 2)} \\
& \text { (3) } \frac{4}{- 1\left(x^{2}+x- 2\right)} \\
& \text { (4) } \frac{x^{2}+ 6 x+ 1}{x(x+ 2)}
\end{aligned}
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Answer
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Step 1:Let's solve this step by step:
Step 2:: Simplify the first fraction \frac{x+ 5}{x+ 2}
- This fraction is already in its simplest form.
Step 3:: Simplify the second fraction \frac{x+ 1}{x^{2}+ 2x}
- Factor the denominator: $$x^{2}+2x = x(x+2)
- The fraction becomes \frac{x+ 1}{x(x+ 2)}
Step 4:: Subtract the fractions
\frac{x+5}{x+2} - \frac{x+1}{x(x+2)}
Step 5:: Find a common denominator
\frac{(x+5)x}{x(x+2)} - \frac{x+1}{x(x+2)}
- Multiply the first fraction by \frac{x}{x}:
Step 6:: Subtract the numerators
= \frac{x^{2}+4x - 1}{x(x+2)}
= \frac{x^{2}+ 5x - x - 1}{x(x+ 2)}
Step 7:: Compare with the given options
- This matches option (1): $$\frac{x^{2}+4x-1}{x(x+2)}
Final Answer
The correct option is (1) \frac{x^{2}+ 4x- 1}{x(x+ 2)}.
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