Mathematical Formulas

Solves $ax^2+bx+c=0$. The discriminant $\Delta = b^2-4ac$ determines root nature: $\Delta>0$ two real roots, $\Delta=0$ one root, $\Delta<0$ complex roots.

Relations between roots and coefficients of $ax^2+bx+c=0$. Generalizes to higher-degree polynomials via elementary symmetric polynomials.

Where $\binom{n}{k} = \dfrac{n!}{k!(n-k)!}$ is the binomial coefficient. Valid for all real $a, b$ and non-negative integer $n$.

Special cases: $a^2-b^2=(a-b)(a+b)$; $a^3-b^3=(a-b)(a^2+ab+b^2)$.

Fundamental properties. Valid for $b>0, b\neq 1$, $x,y>0$.

Converts any logarithm to natural log ($\ln$) or common log ($\log_{10}$).

Core rules for manipulating exponential expressions.

Arithmetic mean is always $\geq$ geometric mean for non-negative reals. Equality holds iff all values are equal.

Common difference $d = a_{k+1} - a_k$. Sum of first $n$ terms.

Common ratio $r = a_{k+1}/a_k$. For $|r|<1$: $S_\infty = \dfrac{a_1}{1-r}$.

$0!=1$ by convention. $P(n,r)$ counts ordered arrangements of $r$ items from $n$.

Number of ways to choose $r$ items from $n$ without regard to order. Pascal's identity: $\binom{n}{r}=\binom{n-1}{r-1}+\binom{n-1}{r}$.

For a right triangle with legs $a, b$ and hypotenuse $c$. Generalizes to $c^2 = a^2+b^2-2ab\cos C$ (Law of Cosines).

Area of a triangle with sides $a, b, c$ using semi-perimeter $s$. No height required.

Area when two sides $a, b$ and their included angle $C$ are known.

$R$ = circumscribed circle radius, $r$ = inscribed circle radius, $A$ = triangle area, $s$ = semi-perimeter.

Distance between two points and midpoint formula in 2D.

Standard forms with center/vertex at $(h,k)$.

Definitions for right triangles. Reciprocals: $\csc = 1/\sin$, $\sec = 1/\cos$, $\cot = 1/\tan$.

Derived from the Pythagorean theorem on the unit circle.

Fundamental expansion used in many derivations.

Note the sign reversal: $+$ on left gives $-$ on right.

Undefined when denominator equals zero.

Special case of sum formulas with $\alpha = \beta = \theta$.

Sign depends on the quadrant of $\theta/2$.

Useful for integration of products of trig functions.

Relates sides to opposite angles. $R$ is the circumradius. Used for AAS/ASA triangle solving.

Generalization of the Pythagorean theorem. Used for SAS/SSS. When $C=90°$, reduces to $c^2=a^2+b^2$.

Applies when limit is in $0/0$ or $\infty/\infty$ indeterminate form. May be applied repeatedly.

Fundamental limits used throughout calculus and analysis.

Constant, power, product, and quotient rules.

For composite functions. Equivalently: $\dfrac{dy}{dx} = \dfrac{dy}{du}\cdot\dfrac{du}{dx}$.

Domain restrictions apply: $|x| \leq 1$ for $\arcsin, \arccos$.

$e^x$ is unique: its own derivative. $a>0$, $a\neq1$.

Differentiate both sides w.r.t. $x$, treating $y$ as a function of $x$, then solve for $dy/dx$.

Derived from the product rule. Choose $u$ using LIATE: Logarithm, Inverse trig, Algebraic, Trig, Exponential.

Reverse of the chain rule. Change limits accordingly for definite integrals.

Expansion of $f$ around point $a$. Maclaurin series is the special case $a=0$.

Valid within radius of convergence.

Only defined when $A$ is $m\times n$ and $B$ is $n\times p$. Not commutative: $AB \neq BA$ in general.

The transpose of a product reverses the order. Symmetric matrix: $A = A^T$.

Exists iff $\det A \neq 0$. $(AB)^{-1} = B^{-1}A^{-1}$. For $2\times 2$: $\begin{pmatrix}a&b\\c&d\end{pmatrix}^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}$.

Cofactor expansion along any row or column. $C_{ij} = (-1)^{i+j}M_{ij}$ where $M_{ij}$ is the minor.

Eigenvalue $\lambda$ and eigenvector $\mathbf{v}\neq\mathbf{0}$. The characteristic polynomial has degree $n$ for an $n\times n$ matrix.

Sum and product of eigenvalues equal trace and determinant respectively.

Scalar result. $\mathbf{a}\perp\mathbf{b}$ iff $\mathbf{a}\cdot\mathbf{b}=0$. Used to find angles and projections.

Vector result perpendicular to both $\mathbf{a}$ and $\mathbf{b}$. Magnitude equals parallelogram area.

Orthogonal projection of $\mathbf{a}$ onto $\mathbf{b}$.

Orthogonalizes a set of linearly independent vectors. Normalize $\mathbf{u}_k/|\mathbf{u}_k|$ for orthonormal basis.

$\sigma^2$ = population variance, $s^2$ = sample variance (Bessel's correction: $n-1$). $\sigma = \sqrt{\sigma^2}$.

Pearson correlation $r$ measures linear association strength and direction.

Relates conditional probabilities. $P(B) = \sum_i P(B|A_i)P(A_i)$ by total probability theorem.

Two events are independent iff their joint probability equals the product of marginal probabilities.

Linearity: $E[aX+b]=aE[X]+b$. For independent $X,Y$: $\text{Var}(X+Y)=\text{Var}(X)+\text{Var}(Y)$.

Sample mean of i.i.d. random variables converges to a normal distribution regardless of the original distribution. Foundation of inferential statistics.

Use $z$ when $\sigma$ known, $t$ when $\sigma$ unknown (with $n-1$ degrees of freedom).

For 95% CI: $z_{0.025} = 1.96$. Width decreases as $n$ increases. Larger $n$ → tighter interval.

Gauss's formula and higher power sums. Note $\sum k^3 = (\sum k)^2$.

Infinite geometric series converges iff $|r| < 1$.

Converges for $|x-a| < R$ (Hadamard formula). Diverges for $|x-a| > R$.

Euler's solution to the Basel problem and other remarkable closed forms.

$a = \text{Re}(z)$, $b = \text{Im}(z)$. $z\bar{z} = |z|^2$. Modulus $|z|$ is the distance from origin.

Multiplication in polar form: $z_1 z_2 = r_1 r_2\, e^{i(\theta_1+\theta_2)}$.

One of the most beautiful equations in mathematics. Setting $\theta=\pi$: $e^{i\pi}+1=0$ (Euler's identity).

Used to find powers and $n$-th roots of complex numbers. Roots: $z_k = r^{1/n}e^{i(\theta+2k\pi)/n}$, $k=0,\ldots,n-1$.

Express trigonometric functions using complex exponentials.

Analog of trig functions. Connection: $\cos(ix) = \cosh x$, $\sin(ix) = i\sinh x$.

Separate variables $x$ and $y$ to opposite sides, then integrate both.

Multiply through by integrating factor $\mu$ to make left side an exact derivative.

Two distinct real roots $r_1,r_2$: $y = C_1 e^{r_1 x}+C_2 e^{r_2 x}$
Repeated root $r$: $y = (C_1+C_2 x)e^{rx}$
Complex roots $\alpha\pm\beta i$: $y = e^{\alpha x}(C_1\cos\beta x+C_2\sin\beta x)$

$W$ is the Wronskian. $y_1, y_2$ are linearly independent solutions to the homogeneous equation.

For integers $a, b > 0$, there exist unique $q$ (quotient) and $r$ (remainder). Foundation for GCD algorithms.

Repeated application reduces to the base case. Also finds $x,y$ with $ax+by=\gcd(a,b)$ (extended GCD).

Linear Diophantine $ax+by=c$ has integer solutions iff $\gcd(a,b) \mid c$.

Every integer $n > 1$ has a unique prime factorization (up to order). Basis for much of number theory.

$p$ is prime. Basis of RSA cryptography. Used for fast modular exponentiation.

Euler's totient $\phi(n)$ counts integers $\leq n$ coprime to $n$. Generalizes Fermat's little theorem.

If $n_1,\ldots,n_k$ are pairwise coprime, the system has a unique solution modulo $N$.

Characterization of prime numbers using factorials. Theoretically elegant but computationally impractical for primality testing.