What is the largest value of k such that the equation 6x - x^2 = k has at least one real solution?
To solve this question, we should use the quadartic equation.
If you don't know what that is, here's a wiki page: https://en.wikipedia.org/wiki/Quadratic_equation
Whether or not a quadratic has a real answer is based on it's discriminant, b^2 - 4ac.
If the discriminant is negative, then the solutions won't work since you can't square root a negative number.
6x - x^2 = k
x^2 - 6x + k = 0
b^2 - 4ac
(-6)^2 - 4(1)(k)
36 - 4k
So, we're looking for the greatest k value where 36 - 4k is non-negative.
The smallest non-negative number is 0.
36 - 4k = 0
36 = 4k
k = 9
Thus, our answer is 9.
I hope this helped. :))))