Descartes’ showed that you can solve quadratics in the
form 
,
where 
by intersecting a circle with a horizontal
line. This is the case when the
horizontal line is 
and when the circle is 
. (Rubinstein, 3)
Proof:
To derive the equation of the circle, start with .
Subtract the 
on both sides: 
Half and square 
to complete the square. 
Simplify and rewrite : 
Add 
on both sides: 
.
Let 
. Then
(Rubinstein, 3)
Why the horizontal line ?
Because we let .
(Allaire and Bradley, 312-313)
See it in action!
Use the applet below to see the x-values of the intersections of the circle and the line. Compare with the x-values of the roots of f(x).
Move p (the coefficient on x in f(x)) and see how it effects the circle.
Move q (the constant of f(x)) and see how it moves the line.
Click the checkbox on f(x) to see the quadradic and its roots.
(Rubinstein, 3)
Click here for a challenge.