Quadratic EquationHard
Question
The poduct of all values of $x$ satisfying the equation $\frac{1}{x^{2} + 2x} + \frac{1}{x^{2} + 6x + 8} + \frac{1}{x^{2} + 10x + 24} = \frac{1}{5} - \frac{1}{x^{2} + 14x + 48}$ is
Options
A.-80
B.40
C.-10
D.-20
Solution
$\frac{1}{2}\left( \frac{1}{x} - \frac{1}{x + 2} + \frac{1}{x + 2} - \frac{1}{x + 4} + \frac{1}{x + 4} - \frac{1}{x + 6} \right) = \frac{1}{5} - \frac{1}{2}\left( \frac{1}{x + 6} - \frac{1}{x + 8} \right)$
$$\begin{matrix} \Rightarrow & \frac{1}{2}\left( \frac{1}{x} - \frac{1}{x + 8} \right) & = \frac{1}{5} \\ \Rightarrow & x^{2} + 8x - 20 & = 0 \\ \Rightarrow & x & = - 10,2 \end{matrix}$$
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