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Solve for x: 1/(x+1)+2/(x+2)=4/(x+4), x ≠ -1, -2, -3 - Mathematics

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Solve for x: `1/(x+1)+2/(x+2)=4/(x+4), `x ≠ -1, -2, -3

 
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Solution

`1/(x+1)+2/(x+2)=4/(x+4)`

L.C.M. of all the denominators is (x + 1)(x + 2)(x + 4)

Multiply throughout by the L.C.M.,we get

(x + 2)(x + 4) + 2(x + 1)(x + 4) = 4(x + 1)(x + 2)

∴ (x + 4)(x + 2 + 2x + 2) = 4(x2 + 3x + 2)

∴ (x + 4)(3x + 4) 4x2 + 12x + 8

∴ 3x2 + 16x + 16 = 4x2 + 12 x 8

∴ x2-4x-8=0

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

`x=(-b+-sqrt(b^2-4ac))/(2a)=(4+-sqrt(16+32))/2=(4+-sqrt48)/2=(4+-4sqrt3)/2`

`:.x=2+-2sqrt3`

Concept: Nature of Roots
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