Question

solve the quadratic equation1/x+1 +2/x+2 =4/x+4

Answer 1

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Answer 2

Answer:

$x = 2±2\sqrt{3}$

Step-by-step explanation:

$\frac{1}{x+1}+\frac{2}{x+2}=\frac{4}{x+4}$

$\implies \frac{x+2+2(x+1)}{(x+1)(x+2)}=\frac{4}{x+4}$

$\implies \frac{x+2+2x+2}{x^{2}+2x+x+2}=\frac{4}{x+4}$

$\implies \frac{3x+4}{x^{2}+3x+2}=\frac{4}{x+4}$

$\implies (3x+4)(x+4)=4(x^{2}+3x+2)$

$\implies 3x^{2}+12x+4x+16=4x^{2}+12x+8$

$\implies 3x^{2}+12x+4x+16-4x^{2}-12x-8=0$

$\implies -x^{2}+4x+8=0$

$\implies x^{2}-4x-8=0$

Compare this equation with ax²+bx+c=0, we get

a=1 , b = -4, c = -8

$Discreminant (D)=b^{2}-4ac\\=(-4)^{2}-4\times 1\times (-8)\\=16+32\\=48$

By Quadratic Formula:

$x = \frac{-b±\sqrt{D}}{2a}$

$\implies x=\frac{-(-4)±\sqrt{48}}{2\times 1}\\=\frac{4±4\sqrt{3}}{2}\\=\frac{2(2±2\sqrt{3}}{2}\\=2±2\sqrt{3}$

Therefore,

$x = 2±2\sqrt{3}$

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