For what value of k does \(x^2 -(k+3)x + (k+2) = 0\) have equal roots?
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For a quadratic \(ax^2+bx+c=0\) to have equal roots, its discriminant must be zero: \(b^2-4ac=0\).
Here \(a=1,\ b=-(k+3),\ c=(k+2)\), so \((k+3)^2 - 4(k+2) = 0\).
Expanding: \(k^2+6k+9-4k-8=0 \Rightarrow k^2+2k+1=0\).
This factors as \((k+1)^2=0\), giving \(k=-1\).
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