Carlyle's Method

Point is the midpoint of line segment , and is the center of the circle passing through points and . Notice that the circle intersects the -axis at and , and that and are the roots of the quadratic function .
INVESTIGATE:
  1. Change the quadratic to . Relocate point so the circle intersects the -axis at the roots of the new quadratic.
  2. Change the quadratic to . Relocate point so the circle intersects the -axis at the roots of the new quadratic.
  3. Change the quadratic to anything of the form . Relocate point so the circle intersects the -axis at the roots of the new quadratic.
  4. Make a conjecture as to where you should place point in order to find the roots of the general quadratic function . Can you prove your conjecture?
Adapted from material of Richard DeCesare.