ALGEBRA
Vieta's Formulas
For \(ax^2+bx+c=0\) with roots \(r,s\):
\(r+s=-\tfrac{b}{a},\quad rs=\tfrac{c}{a}\)
ALGEBRA
AM–GM Inequality
For positive reals \(a_1,\ldots,a_n\):
\(\dfrac{a_1+\cdots+a_n}{n}\ge\sqrt[n]{a_1\cdots a_n}\)
NUMBER THEORY
Units Digit Cycles
Powers of 7 cycle: 7, 9, 3, 1 (period 4). Use \(n\bmod 4\) to find units digit of \(7^n\).
COMBINATORICS
Stars & Bars
Non-neg. integer solutions to \(x_1+\cdots+x_k=n\):
\(\dbinom{n+k-1}{k-1}\)
GEOMETRY
Chord–Distance
If chord has length \(2\ell\) in circle of radius \(r\), distance to center:
\(d=\sqrt{r^2-\ell^2}\)
TRIGONOMETRY
Law of Cosines
\(c^2=a^2+b^2-2ab\cos C\)
LOGARITHMS
Change of Base
\(\log_b a=\dfrac{\log a}{\log b}\)
SEQUENCES
Geometric Series
\(S_\infty=\dfrac{a}{1-r},\quad|r|<1\)
COMPLEX NUMBERS
Modulus of Powers
For \(z=a+bi\): \(|z^n|=|z|^n\)
\(|a+bi|=\sqrt{a^2+b^2}\)
PROBABILITY
Conditional Probability
\(P(A|B)=\dfrac{P(A\cap B)}{P(B)}\)