\(\text{Find all positive values of $c$ so that the inequality $x^2-6x+c<0$ has real solutions for $x$. Express your answer in interval notation. }\)

Guest Jul 18, 2019

#1**+2 **

x^2 - 6x + c < 0

Set this = 0

If this has real solutions, then the discriminant is ≥ 0

So

6^2 - 4c ≥ 0

36 - 4c ≥ 0

36 ≥ 4c

c ≤ 9

When c = 9...the parabola will have its vertex at (3,0)......so....since it turns upward, this is the low point on the graph....so...it is not less than 0 at any point

So.....this implies that the inequality will be < 0 [ i.e., have real solutions for x ] when c = (0, 9)

Here's a graph to show this : https://www.desmos.com/calculator/frqkypfzn8

CPhill Jul 18, 2019