## Basic Math
1. Simple arithmetic:
\(2 + 2 = 4\)
2. Fractions:
\[\frac{1}{2} + \frac{1}{3} = \frac{5}{6}\]
## Algebra
3. Quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
4. Binomial expansion:
\[(x + y)^2 = x^2 + 2xy + y^2\]
## Calculus
5. Derivative notation:
\[\frac{d}{dx} x^n = nx^{n-1}\]
6. Integration:
\[\int_0^1 x^2 dx = \frac{1}{3}\]
7. Taylor series:
\[e^x = \sum_{n=0}^{\infty} \frac{x^n}{n!}\]
## Linear Algebra
8. Matrix multiplication:
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix} =
\begin{pmatrix}
ax + by \\
cx + dy
\end{pmatrix}
\]
9. Determinant:
\[\det\begin{pmatrix}
a & b \\
c & d
\end{pmatrix} = ad - bc\]
## Physics
10. Einstein's mass-energy equivalence:
\[E = mc^2\]
11. Schrödinger equation:
\[i\hbar\frac{\partial}{\partial t}\Psi = \hat{H}\Psi\]
## Invalid input examples
12. Wrong expression is rendered:
\(2++2\)
13. Inline closing delimeter is required:
\(x^2
14. Katex syntax error:
\[\int_0^\infty e^{-x} dx = 1 +}\]