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Find the smallest $k \in Z$ such that the equation

x^{2} - 2 x - (k + 1) = 0

has two distinct solutions.

Hint

The equation has two distinct solutions if and only if

b^{2} - 4 a c > 0

Solution

Since there are two distinct solutions, we have

\begin{aligned} b^{2} - 4 a c & > 0 \\ 4 + 4 (k + 1) & > 0 \\ k + 1 & > - 1 \\ k & > - 2 \end{aligned}

So the smallest integer possible is $k = - 1$ .