College Algebra, Chapter 1, 1.5, Section 1.5, Problem 22

Find all real solutions of the equation $\displaystyle 1 + \frac{2x}{(x + 3)(x + 4)} = \frac{2}{x + 3} + \frac{4}{x + 4}$


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\begin{equation}
\begin{aligned}

1 + \frac{2x}{(x + 3)(x + 4)} =& \frac{2}{x + 3} + \frac{4}{x + 4}
&& \text{Given}
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(x + 3)(x + 4) + 2x =& 2(x + 4) + 4(x + 3)
&& \text{Multiply the LCD } (x + 3)(x + 4)
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x^2 + 7x + 12 + 2x =& 2x + 8 + 4x + 12
&& \text{Expand using FOIL method and Distributive Property}
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x^2 + 3x =& 8
&& \text{Combine like terms}
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x^2 + 3x + \frac{9}{4} =& 8 + \frac{9}{4}
&& \text{Complete the square: add } \left( \frac{3}{2} \right)^2 = \frac{9}{4}
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\left(x + \frac{3}{2} \right)^2 =& \frac{41}{4}
&& \text{Perfect Square}
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x + \frac{3}{2} =& \pm \sqrt{\frac{41}{4}}
&& \text{Take the square root}
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x =& \frac{-3}{2} \pm \frac{\sqrt{41}}{2}
&& \text{Subtract } \frac{3}{2} \text{ and simplify}
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x =& \frac{-3 + \sqrt{41}}{2} \text{ and } x = \frac{-3 - \sqrt{41}}{2}
&& \text{Solve for } x

\end{aligned}
\end{equation}
$