This is a prototype of JavaSketchpad, a World-Wide-Web component of The Geometer's Sketchpad. Copyright ©1990-1998 by Key Curriculum Press, Inc. All rights reserved. Portions of this work were funded by the National Science Foundation (awards DMI 9561674 & 9623018).
Descartes' solution to quadratics
Drag endpoints to change coefficients a and b.
LN = 1/2 a
LM = b
Note how for certain values of a and b the equation has two, one, or zero solutions, which you can see by looking at the intersection with the circle.
OM is the solution to x^2 = ax + b^2
OM = 1/2 a + Sqrt[1/4 a^2 + b^2]
MP is the solution to x^2 = - ax + b^2
MP = 1/2 a - Sqrt[1/4 a^2 +b^2]
MG and MR are solutions to x^2 = ax - b^2
MG = 1/2 a - Sqrt[1/4 a^2 - b^2]
MR = 1/2 a + Sqrt[1/4 a^2 - b^2]
To prove the last two, use right triangle NL'G