Algebra
x² − 5x + 6 = 0→ 2, 3
Galois group of x³ − 2→ S₃
Factor x³ − 6x² + 11x − 6→ (x−1)(x−2)(x−3)
Calculus
d/dx (x³ − 3x² + 2x)→ 3x² − 6x + 2
∫₀³ x² dx→ 9
Critical points of x³ − 3x→ min, max
Number Theory
φ(360)→ 96
Factorize 2520→ 2³·3²·5·7
7¹⁰⁰ mod 13→ fast exp
Geometry
Circumcenter of △(0,0)(4,0)(0,3)→ exact
Area of △(1,1)(4,5)(7,2)→ shoelace
Linear Algebra
Eigenvalues of [[2,1],[1,2]]→ 1, 3
det [[1,2,3],[4,5,6],[7,8,9]]→ 0
Combinatorics
Partitions of 15→ 176
C(20, 7)→ 77,520
Derangements D(8)→ 14,833
Proofs
Prove a² + b² ≥ 2ab→ Proved ∎
Infinitely many primes→ Euclid ∎
Advanced
Points on E/F₃₁: y² = x³+x+1→ Hasse-Weil
Class number of ℚ(√−23)→ h = 3
Is x³−2 solvable by radicals?→ Yes (deg ≤ 4)