Expert: Paul Klarreich - 2/15/2008

Question
I need to solve the inequality x(x-2)>3 by using graph f(x)=x^2-2x-3. Find x intercepts and vertex. I found x intercepts can't seem to get vertex.
Thanks

Answer
Questioner:   Amber
Category:  Advanced Math
Private:  No
 
Subject:  precalc
Question:  I need to solve the inequality x(x-2)>3 by using graph f(x)=x^2-2x-3. Find x intercepts and vertex. I found x intercepts can't seem to get vertex.
Thanks
............................................
Hi, Amber,

If you found the x-intercepts, you are basically done.  You will make these observations:

x(x - 2) > 3  is equivalent to:

x^2 - 2x > 3, which is equivalent to:

x^2 - 2x - 3 > 0, which is equivalent to asking

Where is the graph of f(x) = x^2 - 2x - 3  below the x-axis?

But that is a parabola, matching the form  ax^2 + bx + c
Since  a = 1, positive, it opens UPWARD and its vertex is a minimum.
So the portion BETWEEN its x-intercepts is below the x-axis.
And that portion constitutes the solution set.

No doubt you found  x = -1 and x = 3 as the intercepts, right?

So write   -1 < x < 3  as the solution.

Finding the vertex is irrelevant.  [But if you have to, it is always exactly in between the x-intercepts, isn't it?]