Solve for $x$ .
$\frac{1}{(x - 1) (x - 2)} + \frac{1}{(x - 2) (x - 3)} + \frac{1}{(x - 3) (x - 4)} = \frac{1}{6}$ .

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Solution

$\begin{matrix} \frac{1}{(x - 1) (x - 2)} + \frac{1}{(x - 2) (x - 3)} + \frac{1}{(x - 3) (x - 4)} = \frac{1}{6} \Rightarrow \frac{(x - 3) (x - 4) + (x - 1) (x - 4) + (x - 1) (x - 2)}{(x - 1) (x - 2) (x - 3) (x - 4)} = \frac{1}{6} \Rightarrow 6 (x^{2} - 7 x + 12 + x^{2} - 5 x + 4 + x^{2} - 3 x + 2) - (x^{2} - 3 x + 2) (x^{2} - 7 x + 12) \Rightarrow 6 (3 x^{2} - 15 x + 18) = (x^{4} - 10 x^{3} - 35 x^{2} - 50 x + 24) D i v i s o r = x^{2} - 5 x + 6 Q u o t i e n t = x^{2} - 5 x + 4 R e m a i n d e = 0 \Rightarrow 18 = x^{2} - 5 x + 4 \Rightarrow x^{2} - 5 x - 14 = 0 \Rightarrow x^{2} + 2 x - 7 x - 14 = 0 \Rightarrow x (x + 2) - 7 (x + 2) = 0 \Rightarrow (x - 7) (x + 2) = 0 x = 7 o r - 2 \end{matrix}$

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