What is the largest value of k such that the equation 6x - x^2 = k has at least one real solution?
+308 x^2-6x+k=0
\(x = {-(-6) \pm \sqrt{(-6)^2-4(1)(k)} \over 2(1)}=3\pm1\sqrt{9-k}\)
If k gets any bigger than 9, there both roots will be zero. Hence the largest value k can be such that that equation has at least one real solution is 9.
