1. Substitution of variables can be used to reduce two equations in two unknowns to a single equation in one of them.

   If in a problem the value of a certain variable is neither given nor requested, a reasonable approach is to eliminate that variable.

2. x^a x^b = x^{a + b}
   x^a / x^b = x^{a - b}
   (x^a)^b = x^{a b}
3. If A x² + B x + C = 0, x = (-B ± √ (B² - 4 A C)) / (2 A).
4. e^{ln x} = x
   ln (e^x) = x
5. The graph of y = a f(x + b) is the graph of f(x), dilated in the y direction by the factor a and translated to the left by b.
6. n! ≡ n * (n - 1) * (n -2) * ... * 3 * 2 * 1
7. The number of ways of choosing m items from a set of n items, when order does not matter, is

   n! / (m! (n-m)!)

8. The weighted average of a quantity Q with values in two sub-populations (a and b) is

   <Q> = f_a * Q_a + (1 - f_a) * Q_b,

   where f_a is the fraction of the total population having value Q_a, and the rest of the population has value Q_b.

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©2012, Kenneth R. Koehler. All Rights Reserved. This document may be freely reproduced provided that this copyright notice is included.