all the values of c for the equation: x^2-5x+c>2

Question

We need to find all the values of c for this equation: x^2-5x+c>2 This question was on my exam I didn't know how to get started on the question. Could some one help me out.

Answer 1

You have: x^2 - 5x + c - 2 > 0. The LHS is a parabola with a = 1 > 0. So all you need is the vertex slightly above the x-axis. x-vertex = -b/2a = -(-5)/2(1) = 5/2. So you want f(5/2) > 0. f(5/2) = 25/4 - 25/2 + c - 2 > 0 so c > 2 + 25/4 = 33/4

Answer 2

HINT:

$$x^2-5x=x^2-2\cdot x\cdot\frac52=\left(x-\frac52\right)^2-\left(\frac52\right)^2\ge -\frac{25}4$$ for real $x$