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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

Nathan Axel

#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])#

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Answer 2

Kennedy Adams

To solve the equation ( (\sqrt{x - p}) + (\sqrt{x - q}) = \frac{p}{\sqrt{x - p}} + \frac{q}{\sqrt{x - q}} ), first square both sides of the equation to eliminate the square roots. Then, simplify the resulting equation to solve for ( x ). After solving for ( x ), ensure to check for any extraneous solutions.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.