What is the largest value of k such that the equation 6x - x^2 = k has at least one real solution?

Hi Guest!

Background

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. 

Equation

6x - x^2 = k

x^2 - 6x + k = 0

Discriminant

b^2 - 4ac

(-6)^2 - 4(1)(k)

36 - 4k 

Solving

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

Answer

Thus, our answer is 9. 

I hope this helped. :))))

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