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!
+129850 First one
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
