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Question
`1/(x+1)+2/(x+2)=5/(x+4),x≠-1,-2,-4`
Solution
`1/(x+1)+2/(x+2)=5/(x+4),x≠-1,-2,-4`
⇒`(x+2+2x+2)/((x+1)(x+2))=5/(x+4)`
⇒`(3x+4)/(x^2+3x+2)=5/(x+4)`
⇒`(3x+4) (x+4)=5(x^2+3x+2)`
⇒`3x^2+16x+16=5x^2+15x+10`
⇒`2x^2-x-6=0`
⇒`2x^2-4x-3x-6=0`
⇒`2x(x-2)+3(x-2)=0`
⇒`(x-2)(2x+3)=0`
⇒`3x^2+16x+16=5x^2+15x+10`
⇒`2x^2-x-6=0`
⇒`2x^2-4x+3x-6=0`
⇒`2x(x-2)+3(x-2)=0`
⇒`(x-2) (2x+3)=0`
⇒`x-2=0 or 2x+3=0`
⇒`x=2 or x=-3/2`
Hence, 2 and `-3/2` are the roots of the given equation.
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