Parabola: x-coordinate is negative when it should be positive. I'm teaching myself math. I was working on $y = -x^2 + 8x$ and I used the quadratic equation to determine the x-coordinates. I ended up with $(0,0)$ and $(-8,0)$. The vertex is $(4,16)$ and the y-intercept is $(0,0)$. The thing is, x coordinate of root should be a positive number$=+8,$  otherwise I cannot have a parabola.

Where is my error?
You have that $-x^2 + 8x = -x(x-8)$ so the intersections of the parabola with the $x$-axis are $(0,0)$ and $(8,0)$ as you remarked. You probably made a sign mistake somewhere along your computations.
The thing is, 8 should be a positive number, otherwise I cannot have a parabola.

This makes no sense.  Also, all your calculated points are correct.  It may relieve your confusion for you to know that the parabola is concave (i.e opens downward) and the vertex is above the curve and on the positive side of the y-axis.