Rearranging, this is the equation x^2 - 6x = -k. Completing the square, we add 9 to both sides, and x^2 - 6x + 9 = 9 - k. Factoring the left side, we have (x - 3)^2 = 9 - k. In order for this equation to have at least one real solution, the right-hand side must be nonnegative. Thus, we have 9 - k [tex]\geq[/tex] 0 and k [tex]\leq[/tex] 9. Thus, the largest possible value of k is 9.
