Question

Solve for `x` the equation `(x - p)^(1/2) + (x - q)^(1/2) = p/(x - p)^(1/2) + q/(x - q)^(1/2)` ?

Answer 1

`x = 2/3 (p + q pm sqrt[p^2 - p q + q^2])`

Explanation:

`(x - p)^(1/2) + (x - q)^(1/2) = p/(x - p)^(1/2) + q/(x - q)^(1/2) =`
`p(x-p)^(1/2)/(x-p) + q(x-q)^(1/2)/(x-q)` then

`(x-p)^(1/2)(1-p/(x-p)) = -(x-q)^(1/2)(1-q/(x-q))` or

`((x-p)/(x-q))^(1/2) = -(1-q/(x-q))/(1-p/(x-p))=-((x-p)(x-2q))/((x-q)(x-2p))`

squaring both sides

`(x-p)/(x-q) = ((x-p)^2(x-2q)^2)/((x-q)^2(x-2p)^2)`

and finally

`(x-q)/(x-p) = (x-2q)^2/(x-2p)^2`

resulting in

`3(q-p)x^2+4(p^2-q^2)x+4(pq^2-p^2q)=0`

Solving for `x` we obtain

`x = 2/3 (p + q pm sqrt[p^2 - p q + q^2])`