Question

The roots of the equation $x^2+kx+5 = 0$ differ by $\sqrt{61}$. Find the greatest possible value of $k$.

Answer 1

Let the roots be r1,r2. Then r1+r2 = -k and r1r2 = 5. Use (r1 - r2)^2 = (r1 + r2)^2 - 4r1r2. Given (r1 - r2)^2 = 61, we have

(-k)^2 - 4·5 = 61 ⇒ k^2 - 20 = 61 ⇒ k^2 = 81 ⇒ k = ±9.

The greatest possible value is 9.