This applet explores quadratic equations, linking the algebraic methods
with corresponding geometric interpretations.

The window is set up with sliders for a, b, and c, in the
quadratic equation y=a*x^2+b*x+c. The graph of the function is
also given.

One algebraic approach to the equation is to look for roots using
the quadratic equation. For that we compute the discriminant,
D=b^2-4*a*c, then us that to compute the roots, one red and one
green. Notice that if the discriminant is negative, the equation
has no roots, if the discriminant is zero the equation has one root,
and if the discriminant is positive, the equation has two roots.
If the equation can be solved, we also want to look at the equation in
factored form.

A second algebraic approach uses the method of completing the square
to find the vertex of the parabola. This is given in blue.
If the vertex is the point (h,k), then the equation can also be written
in the form y = a(x-h)^2 + k.

The method of completing the square deserves more attention, so we
look at it in more detail in the next window. Once again, sliders
let us control a, b, and c, and we can see how the method
of completing the square works out for various quadratic polynomials