Friday, January 1, 2010

Solving quadratic equations via geometric construction

We can solve a quadratic equation of the form x2 - sx + p = 0, , using the standard construction tools of compass and straightedge.  The method has been attributed to critic Thomas Carlyle.

Construct the circle in the Cartesian plane with center $\inline&space;\left&space;(&space;\frac{s}{2},\frac{p+1}{2}&space;\right&space;)$ and passing through A(0,1).  By symmetry, the circle also passes through and .

Because the circle has center $\inline&space;\left&space;(&space;\frac{s}{2},\frac{p+1}{2}&space;\right&space;)$  and passes through A(0,1), the equation of the circle is

This reduces to

x2 - sxy2 - (p + 1)y + p = 0

So the x-intercepts of the circle are the solutions to  x2 - sx + p = 0.

Alternate justification:

The segment joining the x-intercepts has a length , hence x1 + x2 = s.

intercepts the arc AX1X2, and  intercepts the opposite arc, hence the two angles are supplementary.  But is also supplementary with , so is congruent to , which in turn impies that  .  Hence

,

which implies that , so x1 x2 = p.

Thus (xx1)(x - x2) = x2 - sx + p, and the solutions to x2 - sx + p = 0 are x1 and x2 .