Question

$\frac{1}{x + 1} + \frac{1}{x + 2} = \frac{4}{x + 4}$
X is not equal to -1,-2,-4
answer: 2 or ± 2√3
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Answer 1

${\bold{\underline{\underline{Answer:}}}}$

$\\{\bold{\therefore x=\frac{-1\pm\sqrt{33}}{4}}}$

${\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\underline \bold{Given : } \\ \implies \frac{1}{x + 1} + \frac{1}{x + 2} = \frac{4}{x + 4} \\ \\ \underline \bold{To \: Find : } \\ \implies value \:of \: x =?$

• According to given question :

$\implies \frac{1}{x + 1} + \frac{1}{x + 2} = \frac{4}{x + 4} \\\\\bold{Taking\:LCM\:in\:LHS} \\ \implies \frac{x + 2 + x + 1}{(x + 1)(x + 2)} = \frac{4}{x + 4} \\ \\ \implies \frac{2x + 3}{ {x}^{2} + 3x + 2 } = \frac{4}{x + 4} \\ \\ \implies (2x + 3)(x + 4) = 4 {x}^{2} + 12x + 8 \\ \\ \implies {2x}^{2} + 8x + 3x + 12 = 4 {x}^{2} + 12x + 8 \\ \\ \implies {4x}^{2} - {2x}^{2} + 12x - 11x + 8 - 12 = 0 \\ \\ \implies {2x}^{2} + x - 4 = 0 \\ \\ \bold{Using \: Quadratic \: formula : } \\ \implies x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} \\ \\ \implies x = \frac{ - 1 \pm \sqrt{ {1}^{2} - 4 \times 2 \times - 4} }{2 \times 2} \\ \\ \implies x = \frac{ - 1 \pm \sqrt{1 + 32} }{4} \\ \\ \bold{\implies x = \frac{ - 1 \pm \sqrt{33} }{4}}$

Answer 2

Answer____

$\bold{x = \frac{ - 1 \pm \sqrt{33} }{4} }$

Step-by-step explanation :

$\implies \frac{1}{x + 1} + \frac{1}{x + 2} = \frac{4}{x + 4} \\ \\ \implies \frac{x + 2 + x + 1}{(x + 1)(x + 2)} = \frac{4}{x + 4} \\ \\ \implies \frac{2x + 3}{ {x}^{2} + 3x + 2 } = \frac{4}{x + 4} \\ \\ \implies (2x + 3)(x + 4) = 4 {x}^{2} + 12x + 8 \\ \\ \implies {2x}^{2} + 8x + 3x + 12 = 4 {x}^{2} + 12x + 8 \\ \\ \implies {4x}^{2} - {2x}^{2} + 12x - 11x + 8 - 12 = 0 \\ \\ \implies {2x}^{2} + x - 4 = 0 \\ \\ \implies x = \frac{ - b \pm \sqrt{ {b}^{2} - 4ac} }{2a} \\ \\ \implies x = \frac{ - 1 \pm \sqrt{ {1}^{2} - 4 \times 2 \times - 4} }{2 \times 2} \\ \\ \implies x = \frac{ - 1 \pm \sqrt{1 + 32} }{4} \\ \\ \bold{\implies x = \frac{ - 1 \pm \sqrt{33} }{4}}$