The roots of the equation x^2+kx+5 differ by sqrt{61}. find the greatest possible value of k.

To figure ut the roots use the quadratic formula

x = [-b +- sqrt(b^2-4ac)]/2a

x = [-k +- sqrt(k^2-4(1)(5)]/2(1)

x = [-k + sqrt(k^2 - 20)]/2 or [-k - sqrt(k^2 - 20)]/2 

So the question says these roots differ by sqrt 61, so let's subtract each

[-k + sqrt(k^2 - 20)]/2 - [-k - sqrt(k^2 - 20)]/2 

well the k's cancel in the beginning and we are left with 2sqrt(k^2 - 20)/2, and the 2 on top and bottom reduce to 

sqrt(k^2 - 20), so this equals sqrt 61

Set equal and solve

sqrt(k^2 - 20) = sqrt(61)

k^2 - 20 = 61

k^2 = 81, so k = +9 or -9

The greatest value therefore is  k = +9.