Question

solve briefly and prove that 1/a+b+x = 1/a+1/b+1/x

Answer 1

i hope this may help you

Answer 2

Quadritatic equation,

We have,

1/a + 1/b + 1/x = 1/(a+b+x)

1/a + 1/b = 1/(a+b+x) - 1/x

{(a+b) / ab} = {[x -(a+b+x)] / x(a+b+x)}

{(a+b) / ab} = {[ x-a-b-x )] / x(a+b+x)}

{(a+b) / ab } = {[x -a-b-x)] / x(a+b+x)}

{ (a+b) / ab} = {[- (a+b) ] / x(a+b+x)}

{( a+b)  / ab} = {[-( a+b )]/x(a+b+x)}

{1 /ab} = {-1/x(a+b+x) }
(By cross multiplication)
-ab = x(a+b+x)

x(a+b+x)+ab = 0

ax+bx+x2 +ab = 0

x2  + x(a+b) + ab = 0 
(By middle term factorization of the equation )
x2 + ax + bx + ab = 0

x(x+a) + b(x+a) = 0

(x+a) {x+b} = 0

(x+a) (x+b) = 0

Therefore,
(x+a) = 0, (x+b) = 0
x= -a, x= -b
So the values of  x are -a and -b.

Now placing the value of x
= -a we get,

LHS = 1/a + 1/b + 1/(-a)

= 1/a + 1/b - 1/a

So ,LHS = 1/b

Now RHS = 1/(a + b + x)

= 1/(a + b - a)

= 1/b = LHS

Similarly it could be shown by replacing the value of x by -b.

That's it
Hope it helped (≧▽≦)