find x in term of a and b 1/x+a+b = 1/x+1/a+1b
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1/ [a+b+x] = 1/a+ 1/b + 1/x
( 1 / [a+b+x] ) - (1 / x ) = (1/a) + (1/b)
=> { x - [a+b+x] } / ([a+b+x] *x ) = {a+b} / ab
=> - { a+b} / ( [a+b+x] * x ) = { a+b} / ab
=> -1 / ( [a+b+x] * x ) = 1 / ab
Cross Multiply : -ab = [a+b+x] * x
On Simplification : x^2 + (a+b) x + ab = 0
Applying the Splitting the middle term method :
=> x^2 + (a+b) x + ab = 0
=> [x^2 + ax]+ [bx + ab] = 0
So
x ( x + a) + b ( x + a) = 0
=> (x + a ) * ( x + b ) = 0
Therefore : (x+a) = 0 or (x+b) = 0
Now : x = -a or x = -b
The value of x is -a or -b
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