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Special Integrals - 1

Solve x+2x+...

Question

Solve $\frac{x + 2}{x + 1} + \frac{x + 1}{x + 2} = \frac{5}{2}$

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Solution

$\frac{x + 2}{x + 1} + \frac{x + 1}{x + 2} = \frac{5}{2} \Rightarrow \frac{{(x + 2)}^{2} + {(x + 1)}^{2}}{(x + 1) (x + 2)} = \frac{5}{2} \Rightarrow \frac{(x^{2} + 4 + 4 x) + (x^{2} + 2 x + 1)}{x^{2} + 3 x + 2} = \frac{5}{2} \Rightarrow 2 (2 x^{2} + 6 x + 5) = 5 (x^{2} + 3 x + 2) \Rightarrow 4 x^{2} + 12 x + 10 = 5 x^{2} + 15 x + 10 \Rightarrow x^{2} + 3 x = 0 \Rightarrow x (x + 3) = 0 x = 0, - 3.$

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