What is the largest value of k such that the equation 6x-x^2=k has at least one real solution?
By the quadratic formula, the largest possible value of k is 12.
Rewrite as x^2 - 6x + k = 0
This has at least one real solution as long as the discriminant is greater than or equal to zero.
i.e. 6^2 - 4k >= 0
4k <= 36
So largest value of k is 9.