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If the roots of the quadratic equation $x^2-2kx+k^2-1=0$ lie in the interval $(–4, 5)$, how to find the sum of all possible values of $\lfloor {k} \rfloor$?

Attempt:

$ x^2-2kx+k^2-1=0$ $\Rightarrow (x-k)^2=1 $ $\Rightarrow k=x \mp 1$

From this we could say that $k \in (-3,6)$ when $k=x+1$ and $k \in (-5,4)$ when $k=x-1$, but then how to do the rest?

1 Answers 1

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You want the sum of all possible $\lfloor k\rfloor$ such that both roots of $x^2-2kx+k^2-1=0$ lie in the interval $(-4,5)$. You’ve correctly determined that if $r$ is a root of the quadratic, then $r=k\pm 1$. For what values of $k$ are $k-1$ and $k+1$ both in the interval $(-4,5)$? You need to find the $k$ for which $-4 and $-4 Once you’ve solved those inequalities simultaneously, you’ll have an interval of possible values of $k$, and you should have no trouble determining the possible values of $\lfloor k\rfloor$.

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    Yeah,and the answer is my favorite number.2011-11-07