The roots of the equation $x^2+kx+5 = 0$ differ by $\sqrt{61}$. Find the greatest possible value of $k$.
I notice that this is a quadratic equation, so I can use the quadratic formula to find the roots in terms of $k$. The quadratic formula says that the roots are $x = \frac{-k \pm \sqrt{k^2 - 4(1)(5)}}{2(1)}$. Since the roots differ by $\sqrt{61}$, I can set up an equation using the difference of the two roots: $\frac{-k + \sqrt{k^2 - 20}}{2} - \frac{-k - \sqrt{k^2 - 20}}{2} = \sqrt{61}$. I can simplify this equation by combining the fractions and canceling out the $-k$ terms: $\sqrt{k^2 - 20} = \sqrt{61}$. Squaring both sides, I get $k^2 - 20 = 61$, or $k^2 = 81$. Taking the square root of both sides, I get $k = \pm 9$. The question asks for the greatest possible value of $k$, so I choose the positive root: $k = 9$. 