The quadratic formula solves $ax^2+bx+c=0$ for $x$ giving $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ The binomial theorem expands $(x+y)^n$ as $$\sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$$ A geometric series with ratio $r$ sums to $$\sum_{k=0}^{\infty} ar^k = \frac{a}{1-r}$$ for $|r| < 1$ The nth root of unity is $\sqrt[n]{1}$ and $x^{1/n} = \sqrt[n]{x}$
The derivative of $f(x) = x^n$ is $f'(x) = nx^{n-1}$ by the power rule. The limit definition of the derivative is $$\frac{d}{dx}f(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$$ The fundamental theorem of calculus states $$\int_a^b f(x)\,dx = F(b) - F(a)$$ where $F'(x) = f(x)$ Integration by parts: $$\int u\,dv = uv - \int v\,du$$ A Taylor series expands $f(x)$ about $x=0$ as $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n$$ The gradient of a scalar field is $$\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\right)$$ and its divergence is $\nabla \cdot \mathbf{F}$
The Pythagorean identity is $$\sin^2\theta + \cos^2\theta = 1$$ The angle sum formula gives $$\sin(\alpha+\beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta$$ and $\tan\theta = \frac{\sin\theta}{\cos\theta}$ Euler's formula connects $e^{i\theta} = \cos\theta + i\sin\theta$ trigonometry to complex exponentials, and $e^{i\pi} + 1 = 0$ is the special case at $\theta = \pi$
A 2x2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ has determinant $\det A = ad-bc$ and, when invertible, $$A^{-1} = \frac{1}{\det A}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$$ Matrix multiplication is defined by $$(AB)_{ij} = \sum_{k} A_{ik}B_{kj}$$ A vector norm is written $$\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2}$$ and the dot product $\mathbf{u}\cdot\mathbf{v} = \sum_i u_i v_i$ An eigenvector satisfies $A\mathbf{v} = \lambda \mathbf{v}$ for eigenvalue $\lambda$
For sets $A$ and $B$ the union and intersection are $A \cup B$ and $A \cap B$ De Morgan's laws state $$\overline{A \cup B} = \overline{A} \cap \overline{B}$$ A function $f: A \to B$ is injective if $$\forall x,y \in A,\ f(x)=f(y) \implies x=y$$ We write $x \in \mathbb{R}$ , $n \in \mathbb{N}$ and $z \in \mathbb{C}$ for real, natural, and complex numbers respectively, and $\emptyset \subset A \subseteq \mathbb{R}$ for the empty set.
The normal distribution has density $$f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$ with mean $\mu$ and variance $\sigma^2$ Bayes' theorem states $$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$ The expected value of a discrete variable is $\mathbb{E}[X] = \sum_i x_i P(x_i)$ and its variance $\mathrm{Var}(X) = \mathbb{E}[X^2] - \mathbb{E}[X]^2$
The absolute value function is $$f(x) = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases}$$ A sequence converges if $$\forall \epsilon > 0,\ \exists N,\ \forall n > N,\ |a_n - L| < \epsilon$$ We write $\lim_{n \to \infty} a_n = L$ for the limit, and $a_n \to L$ as $n \to \infty$ equivalently.