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?

Adapted from material of Richard DeCesare.

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