math_math

Quadratics

x² − 5x + 6 = 0→ 2, 3

x² − 7x + 10 = 0→ 2, 5

x² + x − 12 = 0→ −4, 3

2x² − 7x + 3 = 0→ 1/2, 3

x² − 9 = 0→ ±3

3x² − 10x + 3 = 0→ 1/3, 3

x² − 6x + 9 = 0→ 3 (double)

x² + 4x + 4 = 0→ −2 (double)

4x² − 4x + 1 = 0→ 1/2 (double)

Complex roots

x² + 1 = 0→ ±i

x² + 2x + 5 = 0→ −1 ± 2i

x² − 4x + 13 = 0→ 2 ± 3i

Cubics & Quartics

x³ − 6x² + 11x − 6 = 0→ 1, 2, 3

x⁴ − 5x² + 4 = 0→ ±1, ±2

x³ + 6x² + 11x + 6 = 0→ −1, −2, −3

x³ − 7x + 6 = 0→ 1, 2, −3

x³ − x = 0→ 0, ±1

x⁴ − 10x² + 9 = 0→ ±1, ±3

x⁴ − 13x² + 36 = 0→ ±2, ±3

Factoring

Factor x² − 5x + 6→ (x−2)(x−3)

Factor x³ − 27→ (x−3)(x²+3x+9)

Factor x³ − 6x² + 11x − 6→ (x−1)(x−2)(x−3)

Factor x² − 9→ (x+3)(x−3)

Factor x² + 4x + 4→ (x+2)²

Vieta's formulas

Σ roots of x² − 7x + 12→ 7

Π roots of x² − 7x + 12→ 12

Σ roots of 2x² − 10x + 8→ 5

Π roots of 2x² − 10x + 8→ 4

Complex arithmetic

(2+3i)(4−i)→ 11+10i

(1+i)⁴→ −4

(3+4i)(3−4i)→ 25

(2+i)³→ 2+11i

Derivatives

d/dx (x³)→ 3x²

d/dx (x⁴ + 3x² − 2x + 1)→ 4x³ + 6x − 2

d/dx (5x³ − 2x² + x)→ 15x² − 4x + 1

d/dx (x⁵ − x³ + x)→ 5x⁴ − 3x² + 1

d/dx (x⁶)→ 6x⁵

d/dx (x¹⁰)→ 10x⁹

d/dx (7x³ − 4x + 9)→ 21x² − 4

d/dx (x² + 1)→ 2x

d/dx (x⁸)→ 8x⁷

d/dx (4x³ + x)→ 12x² + 1

d/dx (x⁷ − 3x⁵ + 2x³)→ 7x⁶ − 15x⁴ + 6x²

d/dx (10x³ − 6x² + x − 7)→ 30x² − 12x + 1

Definite integrals

∫₀¹ x² dx→ 1/3

∫₀² x³ dx→ 4

∫₁⁵ x dx→ 12

∫₋₁¹ x² dx→ 2/3

∫₀³ 3x² dx→ 27

∫₀¹ x⁴ dx→ 1/5

∫₀⁴ 2x dx→ 16

∫₀² (x²+x) dx→ 14/3

∫₀¹ x⁵ dx→ 1/6

∫₋₁¹ x³ dx→ 0

∫₀³ x² dx→ 9

∫₀² 5x⁴ dx→ 32

∫₀² 4x³ dx→ 16

∫₁⁴ (x²+2x) dx→ 36

Critical points

Extrema of x³ − 3x→ max(−1,2), min(1,−2)

Extrema of x³ − 12x→ max(−2,16), min(2,−16)

Extrema of x² − 6x + 5→ min(3,−4)

Extrema of x³ − 3x²→ max(0,0), min(2,−4)

Extrema of 2x³ − 9x² + 12x→ max(1,5), min(2,4)

Extrema of x³ + 3x² − 9x→ max(−3,27), min(1,−5)

Euler totient

φ(12)→ 4

φ(100)→ 40

φ(36)→ 12

φ(60)→ 16

φ(1000)→ 400

φ(128)→ 64

φ(7)→ 6

Prime factorization

Factorize 360→ 2³·3²·5

Factorize 2520→ 2³·3²·5·7

Factorize 84→ 2²·3·7

Factorize 1000→ 2³·5³

Factorize 72→ 2³·3²

GCD

gcd(12, 18)→ 6

gcd(100, 75)→ 25

gcd(48, 64)→ 16

gcd(144, 60)→ 12

gcd(252, 198)→ 18

Modular arithmetic

7¹⁰⁰ mod 13→ 9

2¹⁰ mod 1000→ 24

3²⁰ mod 7→ 2

2¹⁰⁰ mod 17→ 16

13⁵⁰ mod 17→ 16

Fibonacci & Divisors

F₁₀→ 55

Divisors of 24→ {1,2,3,4,6,8,12,24}

F₁₅→ 610

F₂₀→ 6,765

F₂₅→ 75,025

Divisors of 36→ 9 divisors

Divisors of 60→ 12 divisors

Divisors of 100→ 9 divisors

Triangle area

Area △(0,0)(3,0)(0,4)→ 6

Area △(0,0)(5,0)(0,12)→ 30

Area △(1,1)(4,1)(1,5)→ 6

Area △(2,3)(5,7)(8,1)→ 15

Area △(0,0)(1,0)(0,1)→ 1/2

Area △(0,0)(10,0)(5,8)→ 40

Area △(0,0)(8,0)(4,6)→ 24

Area △(−1,−1)(3,2)(1,5)→ 9

Distance

dist (0,0)↔(3,4)→ d² = 25

dist (0,0)↔(5,12)→ d² = 169

dist (1,1)↔(4,5)→ d² = 25

dist (0,0)↔(8,15)→ d² = 289

dist (0,0)↔(6,8)→ d² = 100

dist (3,4)↔(6,8)→ d² = 25

Triangle centers

Circumcenter △(0,0)(4,0)(0,3)→ (2, 3/2)

Centroid △(0,0)(3,0)(0,6)→ (1, 2)

Circumcenter △(0,0)(6,0)(0,8)→ (3, 4)

Circumcenter △(0,0)(10,0)(0,10)→ (5, 5)

Collinear? (0,0)(1,1)(2,2)→ Yes

Collinear? (0,0)(1,2)(3,6)→ Yes

Collinear? (0,0)(1,1)(2,3)→ No

Collinear? (1,1)(2,3)(3,5)→ Yes

Orthocenter △(0,0)(4,0)(0,3)→ (0, 0)

Orthocenter △(0,0)(6,0)(3,4)→ (3, 9/4)

Centroid △(0,0)(6,0)(0,9)→ (2, 3)

Centroid △(0,0)(12,0)(6,9)→ (6, 3)

Incenter △(0,0)(4,0)(0,3)→ (1, 1)

Incenter △(0,0)(6,0)(0,8)→ (2, 2)

Incenter △(0,0)(8,0)(0,6)→ (2, 2)

Incenter △(0,0)(3,0)(0,4)→ (1, 1)

Determinants

det [1,2; 3,4]→ −2

det [2,5; 1,3]→ 1

det 3×3 [1..9]→ 0

det [2,1,0; 1,3,2; 0,1,4]→ 16

det [5,3; 2,4]→ 14

det [4,6; 3,8]→ 14

det [3,7; 1,4]→ 5

det diag(1,2,3)→ 6

det [1,2,3; 0,1,4; 5,6,0]→ 1

det [1,1,1; 1,2,3; 1,4,9]→ 2

Eigenvalues

λ of [2,1; 1,2]→ 1, 3

λ of [4,1; 2,3]→ 2, 5

λ of diag(3,5)→ 3, 5

λ of [5,2; 2,5]→ 3, 7

λ of [6,−1; 2,3]→ 4, 5

λ of diag(1,2,3)→ 1, 2, 3

Inverse, rank & trace

[1,2; 3,4]⁻¹→ [−2,1; 3/2,−1/2]

rank [1,2; 3,4]→ 2

[2,1; 5,3]⁻¹→ [3,−1; −5,2]

[3,1; 5,2]⁻¹→ [2,−1; −5,3]

[1,1; 0,1]⁻¹→ [1,−1; 0,1]

rank 3×3 [1..9]→ 2

rank [1,2; 2,4]→ 1

rank I₃→ 3

tr [1,2; 3,4]→ 5

tr [5,3; 2,7]→ 12

tr 3×3 [1..9]→ 15

tr diag(10,20,30)+…→ 60

[1,2; 3,4]ᵀ→ [1,3; 2,4]

[1,2,3; 4,5,6]ᵀ→ 3×2 matrix

Combinations

C(10,3)→ 120

C(5,2)→ 10

C(8,4)→ 70

C(20,7)→ 77,520

C(100,2)→ 4,950

C(15,7)→ 6,435

C(6,0)→ 1

C(9,4)→ 126

C(50,3)→ 19,600

Permutations

P(10,3)→ 720

P(8,4)→ 1,680

P(7,3)→ 210

P(6,6) = 6!→ 720

P(12,4)→ 11,880

Derangements & sequences

D(4)→ 9

Cat(5)→ 42

D(5)→ 44

D(6)→ 265

D(8)→ 14,833

D(10)→ 1,334,961

p(10)→ 42

p(15)→ 176

p(20)→ 627

p(25)→ 1,958

Cat(7)→ 429

Cat(10)→ 16,796

S(5,3)→ 25

S(6,2)→ 31

S(7,4)→ 350

S(8,3)→ 966

Inequalities

a² + b² ≥ 2ab→ Proved ∎

(a+b)/2 ≥ √(ab) (AM-GM)→ Proved ∎

(a²+b²)(c²+d²) ≥ (ac+bd)²→ Cauchy-Schwarz ∎

(a − b)² ≥ 0→ Proved ∎

a² + b² + c² ≥ ab + bc + ac→ Proved ∎

a⁴ + b⁴ ≥ a²b²→ Proved ∎

a² + 1 ≥ 2a→ Proved ∎

a² − a + 1 ≥ 0→ Proved ∎

a³ + b³ ≥ a²b + ab²→ Proved ∎

2(a² + b²) ≥ (a+b)²→ Proved ∎

a⁴ + 1 ≥ 2a²→ Proved ∎

a⁶ + b⁶ ≥ a³b³→ Proved ∎

Number theory

n² ≥ 2n − 1→ Proved ∎

Infinitely many primes→ Euclid ∎

Elliptic curves

#E(F₃₁): y²=x³+x+1→ 33

#E(F₁₁): y²=x³+1→ 12

#E(F₅): y²=x³+x+1→ 9

#E(F₁₃): y²=x³+2x+3→ 18

Class numbers

h(ℚ(√−23))→ 3

h(ℚ(√−163))→ 1

h(ℚ(√−71))→ 7

Galois theory

Gal(x³−2)→ S₃

Gal(x²−2)→ Z/2Z

Gal(x³−3x+1)→ A₃

Gal(x³+x+1)→ S₃

Solvability

x⁴−2 solvable?→ Yes

x⁵−x−1 solvable?→ No

x³−2 solvable?→ Yes