if (1/x) + (1/x-a) + (1/x-b)=0.

prove that one root lies between a/3 and 2a/3 , and other lies between b/3 and 2b/3

PLS HELP WITH EXPLANATION..ANSWER WILL SURELY BE APPROVED!!

Hello Student,
Please find answer to your question below
[{1}/{x} +{1}/{x-a} +{1}/{x-b} = 0]
[{1}/{x-a} {1}/{x-b} ={-1}/{x}]
[x(2x-a-b) = -(x^{2}-(a+b)x+ab)]
[3x^2 - 2(a+b)x+ab = 0]
[x = {2(a+b)sqrt{4(a+b)^2-12ab}}/{6}]
[x = {(a+b)sqrt{(a+b)^2-3ab}}/{3}]
From here it is clear that one root lie b/w a/3 & 2a/3 and other b/w b/3 & 2b/3.