Determine the number of x-intercepts that appear on a graph of each function. f (x) = (x - 6) 2 (x + 2) 2

Short answer There are 2 places where the graph touches the x axis just as you said.

I take this to mean f (x) = (x - 6) ^2 * (x + 2) ^2

There are two places where the graph touches the x axis but that is not an intercept.

When a graph just touches the x axis, that counts as a root. As the graph shows, these 2 places are

x = 6 and

x = - 2 is where they touch.

If you have a graphing calculator, you can see what this looks like for yourself. The window settings are

x min = - 3

x max = 7

y min = 0

y max = 250.

There is a local maximum between x = - 2 and x = 6.

So the answer, once again is at x = - 2 and x = 6 and that makes 2 roots for the equation.