Could I please have some help on a few homework questions on quadratics? I'm having some trouble finding the answers.
1. The solution to the inequality \(y = -x^2 + ax + b \le 0 \) is \((-\infty,-3] \cup [5,\infty).\) Find the vertex of the parabola y = -x^2 + ax + b.
2. Find the largest constant C so that \(x^2 + y^2 + xy + 1 \ge C(x + y)\) for all real numbers x and y.
3. Find all values of k so that the graphs of x^2 + y^2 = 4 + 12x + 6y and x^2 + y^2 = k + 4x + 12y intersect.
For the first question, I tried to write out the quadratic equation. For the second, I tried combining terms but I am not sure what to do with C. For the last, I tried making the two equations equal one another, but I'm not sure how to find all the values.
Any help will be appreciated, and thank you very much!
The roots are x = - 3 and x = 5
So....the equation of the parabola is y = - ( x - 5) (x + 3)
y = -x^2 + 2x - 15
Because of symmetry....the x coordinate of the vertex is x = [ -3 + 5] / 2 = 2/2 = 1
So....to find the y coordinate we have
y = -(1)^2 + 2(1) + 15
y = -1 + 2 + 15
y = 16
So...the vertex is (1, 16)
Here's a graph : https://www.desmos.com/calculator/5szp8sqydj
Thank you very much! Do you have any idea on how to approach the other two problems and solve them?