Question

1/X+m+n = 1/X + 1/m + 1/n

Answer 1

Given equation : $\bold{\dfrac{1}{X+m+n}=\dfrac{1}{X}+\dfrac{1}{m}+\dfrac{1}{n}}$

         Solution -

$\implies \mathsf{\dfrac{1}{X+m+n}-\dfrac{1}{X} = \dfrac{1}{m}+\dfrac{1}{n}}\\\\\\\\\implies\mathsf{ \dfrac{X-(X+m+n)}{X(X+m+n)} = \dfrac{1}{m}+\dfrac{1}{n}}\\\\\\\\\implies\mathsf{\dfrac{X^2 - X^2 - m - n }{X(X+m+n)}}\\\\\\\\\implies\mathsf{\dfrac{-(m+n)}{X(X+m+n)} = \dfrac{n+m}{mn}} \\\\\\\\\implies\mathsf{ \dfrac{-1}{X^2 + mX + nX } = \dfrac{1}{mn}}\\\\\\\\\implies \mathsf{X^2 + mX + nX  = -mn }$

$\implies \mathsf{X^2+mX+nX+mn= 0}\\\\\\\implies \mathsf{X(X+m)+n(X+m)}=0\\\\\\\implies \mathsf{(X+m)(X+n)}=0 \\\\\\\therefore \mathsf{X + m = 0 \:\:\:Or\:\:\:X+n=0}$

X = - m   or  X = - n

Therefore the value of X is - m or - n .