Find k such that \(x^2 -2(k+1)x + (k^2+1)=0\) has distinct real roots.
k>0
For distinct real roots, the discriminant must be strictly positive: \(b^2-4ac>0\).
Here \(a=1,\ b=-2(k+1),\ c=(k^2+1)\), so \(4(k+1)^2 - 4(k^2+1) > 0\).
Dividing by 4: \((k+1)^2 - (k^2+1) > 0 \Rightarrow k^2+2k+1-k^2-1>0 \Rightarrow 2k>0\).
This simplifies to \(k>0\).
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