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Addition of Rational Expressions

2x+6/x2+3x-x+...

Question

$\frac{2 x + 6}{x^{2} + 3 x} - \frac{x + 3}{x^{2} + x}$
Choose the correct option for simplified form of the given expression.

- \frac{1}{x}

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\frac{x + x}{2 x}

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\frac{x - 1}{x (x + 1)}

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\frac{x + 9}{2 x^{2} + 4 x}

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Solution

The correct option is C $\frac{x - 1}{x (x + 1)}$

Given, $\frac{2 x + 6}{x^{2} + 3 x} - \frac{x + 3}{x^{2} + x}$

Take out the common terms, we get

$=$ $\frac{x + 3}{x} [\frac{2}{x + 3} - \frac{1}{x + 1}]$

$=$ $\frac{x + 3}{x} [\frac{2 (x + 1) - (x + 3)}{(x + 3) (x + 1)}]$

$=$ $\frac{2 x + 2 - x - 3}{x (x + 1)}$

$=$ $\frac{x - 1}{x (x + 1)}$

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\frac{2 x - 1}{x + 3} + \frac{1 - 2 x}{x - 3} = \frac{4 - 3 x}{x^{2} - 9}

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