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

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Answer 1 · 2016-09-12T19:14:25

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

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

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

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