MHB Week #104: Solving Equation with Real Parameters - March 24, 2014

[anemone]

Solve the equation $\sqrt{x+k}+\sqrt{x+m}+\sqrt{x+n}=\sqrt{x+k+m-n}$ where $k,\,m,\,n$ are real parameters.

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[anemone]

Re: Problem of the week #104 -March 24th, 2014

Congratulations to the following members for their correct solutions::)

1. mente oscura
2. lfdahl

Solution from lfdahl:

Spoiler

\[\sqrt{x+k}+\sqrt{x+m}+\sqrt{x+n}=\sqrt{x+k+m-n} \\\\ \Leftrightarrow (\sqrt{x+k}+\sqrt{x+m})^2=(\sqrt{x+k+m-n}-\sqrt{x+n})^2\\\\ \Leftrightarrow 2x+k+m+2\sqrt{(x+k)(x+m)} = 2x+k+m-n+n-2\sqrt{(x+n)(x-n+k+m)}\\\\ \Leftrightarrow \sqrt{(x+k)(x+m)} = -\sqrt{(x+n)(x-n+k+m)} \\\\ \Rightarrow (x+k)(x+m) = (x+n)(x-n+k+m) \Rightarrow (n-k)(n-m)=0 \\\\ (a). \;\;\;n = k\neq m \Rightarrow x = -n =-k \\\\ (b).\;\;\;n = m \neq k\Rightarrow x = -n=-m \\\\ (c).\;\;\;n=k=m \Rightarrow x = -n=-m=-k\]