Online Course Discussion Forum
MC-III Algebra 5.25
I expanded the original equation and got xy+x(sqrt(y^2-2018)+y(sqrt(x^2-2018))+sqrt(x^2y^2-2018(x^2+y^2)+2018^2)=2018. Just messing around I plugged in y=-x. This gives -2018=2018. I don't know what to do from here. I don't see a clear way to clear the square roots or get to our target equation.
If you instead plugged in $x=y$ you would get $2018=2018$, so this is a hint that maybe $x=y$. Note this, however, is not a justification that $x=y$!
As a hint for deriving $x=y$, note our equation implies$$x+\sqrt{x^2-2018} = \frac{2018}{y+\sqrt{y^2-2018}}.$$How can we continue simplifying here? As a hint, how would you simplify an expression like $\dfrac{1}{A+\sqrt{B}}$ and write it in simplest radical form?
Then can we write a similar equation for $y + \sqrt{y^2-2018}$?
Hope this helps!
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