Theorem 1 (Quadratic Formula) $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

Proof. by $(a + b)^{2} = a^{2} + 2 a b + b^{2}$ $\begin{aligned} a x^{2} + b x + c = 0 (a \neq 0) \\ x^{2} + \frac{b}{a} x + \frac{c}{a} = 0 \\ (x + \frac{b}{2 a})^{2} = - \frac{c}{a} + \frac{b^{2}}{4 a^{2}} \\ (x + \frac{b}{2 a})^{2} = \frac{- 4 a c + b^{2}}{4 a^{2}} \\ x + \frac{b}{2 a} = \pm \sqrt{\frac{b^{2} - 4 a c}{4 a^{2}}} \\ x = \pm \sqrt{\frac{b^{2} - 4 a c}{4 a^{2}}} - \frac{b}{2 a} \\ x = \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a} - \frac{b}{2 a} \\ x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{aligned}$

Definition 1 (Metric Space) A metric space is a set $M$ with a function $d : M \times M \to R$ such that:

$d (x, y) \geq 0$ (Non-negativity)
$d (x, y) = 0$ if and only if $x = y$
$d (x, y) = d (y, x)$ (Symmetry)
$d (x, z) \leq d (x, y) + d (y, z)$ (Triangle inequality)

Definition 2 (The limit of a sequence) Suppose $a_{n}$ is a convergent sequence, and $L \in R$ is the limit

$\forall ϵ > 0, \exists N \in N, \forall n \geq N, | a_{n} - L | < ϵ$

Note: the order of the quantifier is $\forall$ , $\exists$ , $\forall$

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Fundamental Theorem of Calculus

$\int_{a}^{b} f^{'} (x) d x = f (b) - f (a)$

Thm: Multiple Lines Theorem

$\begin{matrix} (1) & \begin{aligned} a x^{2} + b x + c = 0 \\ x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} \end{aligned} \end{matrix}$

Tip 1

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Theorem 2 This is a theorem example for testing reference

Figure 1 Tip 1 Definition 1

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Google

https://quarto.org/docs/get-started/

A\B	B1	B2
A1	A1 x B1	A1 x B2
A2	A2 x B1	A2 x B2

Footnotes

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