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Think of two numbers. Tell me their sum. Tell me their product. The two numbers are ....

A fun and practical application of the quadratic formula, for the 17b/19b/27s solver family.

SUMIS:IF(S(N1):(SUM+SQRT(SQ(SUM)-4$$\times$$PROD))$$\div$$2-N1:(SUM-SQRT(SQ(SUM)-4$$\times$$PROD))$$\div$$2-N2)
In terms of the mean M and product P of two numbers, the quadratic formula becomes x = M +/- √(M^2 - P). I think that that's impressive.
(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
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