# 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:

- Change the quadratic to
. Relocate point so the circle intersects the -axis at the roots of the new quadratic. - Change the quadratic to
. Relocate point so the circle intersects the -axis at the roots of the new quadratic. - Change the quadratic to anything of the form
. Relocate point so the circle intersects the -axis at the roots of the new quadratic. - 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?