How to impress your math students

[Thomas Klemm]

(06-29-2014 01:30 PM)Don Shepherd Wrote:  Think of two numbers. Tell me their sum. Tell me their product. The two numbers are ....

Or tell me their arithmetic and geometric mean. Use the geometric mean theorem and construct a triangle with radius \(r=\frac{p+q}{2}\) and altitude \(h=\sqrt{pq}\).

Then let them figure out how to calculate p and q:
\[p = r + \sqrt{r^2 - h^2}\]
\[q = r - \sqrt{r^2 - h^2}\]

From this it's easy to find the quadratic formula.

Cheers
Thomas