BdMO National Secondary 2010#3
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- Posts:1007
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Solve for real $x$:$\dfrac {|x^2-1|}{x-2}=x$
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- Posts:1007
- Joined:Sat Dec 09, 2017 1:32 pm
Re: BdMO National Secondary 2010#3
$\dfrac {|x^2-1|}{x-2}=x \Rightarrow |x^2-1|=x^2-2x $
$x^2-2x=x^2+1\Rightarrow 2x=1\Rightarrow x=\dfrac 12$
Or,
$x^2-2x=-x^2+1\Rightarrow 2x^2-2x-1=0\Rightarrow x=\dfrac {1+\sqrt 3}{2},\dfrac {1-\sqrt 3}{2}$
From these $3$ values only $\dfrac {1-\sqrt 3}{2}$ is the answer for real $x$.
$x^2-2x=x^2+1\Rightarrow 2x=1\Rightarrow x=\dfrac 12$
Or,
$x^2-2x=-x^2+1\Rightarrow 2x^2-2x-1=0\Rightarrow x=\dfrac {1+\sqrt 3}{2},\dfrac {1-\sqrt 3}{2}$
From these $3$ values only $\dfrac {1-\sqrt 3}{2}$ is the answer for real $x$.
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- Posts:1007
- Joined:Sat Dec 09, 2017 1:32 pm
Re: BdMO National Secondary 2010#3
This is German National Olympiad 2006 2nd round Problem 1.