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Factors of Algebraic Expressions and Factorisation

solve: 1x -...

Question

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

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Solution

$\begin{matrix} \frac{1}{(x - 1) (x + 2)} + \frac{1}{(x - 2) (x - 3)} = \frac{2}{3} \frac{(x - 2) (x - 3) + (x + 2) (x - 1)}{(x - 1) (x - 2) (x + 2) (x - 3)} = \frac{2}{3} \frac{x^{2} - 3 x - 2 x + 6 + x^{2} - 2 - x - 2}{(x^{2} - x - 4) (x^{2} - 5 x + 6)} = \frac{2}{3} 6 x^{2} - 18 x + 6 = 2 (x^{2} - x - 4) (x^{2} - 5 x + 6) 6 x^{2} - 18 x + 6 = 2 (x^{4} - 5 x^{3} + 6 x^{2} - x^{3} + 5 x^{2} - 6 x - 4 x^{2} + 20 x - 24) 6 x^{2} - 18 x + 6 = 2 (x^{4} - 6 x^{3} + 7 x^{2} + 14 x - 24) 6 x^{2} - 18 x + 6 = 2 x^{4} - 12 x^{3} + 14 x^{2} + 28 x - 48 2 x^{4} - 12 x^{3} + 14 x^{2} + 28 x - 48 - 6 x^{2} + 18 x - 6 = 0 2 x^{4} - 12 x^{3} + 8 x^{2} + 38 x - 54 = 0 \end{matrix}$

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