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?