x²
Algebra Key identities, formulas, and rules for solving equations, working with polynomials, sequences, and linear/quadratic functions.
Quadratic Formula
Core
General form
ax² + bx + c = 0
Solution
-b ± √(b² - 4ac) x = ────────────────── 2a
Discriminant: b²−4ac > 0 → 2 real roots; = 0 → 1 repeated root; < 0 → complex roots.
Factoring Identities
Algebra
Difference of squares
a² - b² = (a + b)(a - b)
Perfect square
(a ± b)² = a² ± 2ab + b²
Sum of cubes
a³ + b³ = (a+b)(a² - ab + b²)
Difference of cubes
a³ - b³ = (a-b)(a² + ab + b²)
Lines & Distance
Coordinate
Slope-intercept
y = mx + b
Point-slope
y - y₁ = m(x - x₁)
Slope from two points
y₂ - y₁ m = ───────── x₂ - x₁
Distance
d = √[(x₂-x₁)² + (y₂-y₁)²]
Midpoint
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Exponent Rules
Core
aᵐ · aⁿ = aᵐ⁺ⁿ aᵐ / aⁿ = aᵐ⁻ⁿ (aᵐ)ⁿ = aᵐⁿ (ab)ⁿ = aⁿbⁿ a⁻ⁿ = 1 / aⁿ a⁰ = 1 (a ≠ 0) a^(m/n) = ⁿ√(aᵐ)
Logarithm Rules
Core
logb(xy) = logb(x) + logb(y) logb(x/y) = logb(x) - logb(y) logb(xⁿ) = n · logb(x) logb(b) = 1 logb(1) = 0 log(x) logb(x) = ──────── (change of base) log(b)
ln x = logₑ x  |  log x = log₁₀ x
Sequences & Series
Series
Arithmetic — nth term
aₙ = a₁ + (n − 1)d
Arithmetic — sum
n(a₁ + aₙ) Sₙ = ────────── 2
Geometric — nth term
aₙ = a₁ · rⁿ⁻¹
Geometric — sum
a₁(1 − rⁿ) Sₙ = ────────── (r ≠ 1) 1 − r
Infinite geometric (|r| < 1)
S = a₁ / (1 − r)
Absolute Value & Inequalities
Algebra
|x| = a → x = a or x = −a |x| < a → −a < x < a |x| > a → x < −a or x > a
Flip the inequality sign when multiplying or dividing by a negative number.
Binomial Theorem
Expansion
n (a+b)ⁿ = Σ C(n,k) · aⁿ⁻ᵏ · bᵏ k=0 n! C(n,k) = ──────── k!(n−k)!
First few rows of Pascal's triangle: 1 | 1 1 | 1 2 1 | 1 3 3 1
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Geometry Area, perimeter, volume, surface area, and angle relationships for 2-D and 3-D shapes, plus trigonometry laws.
2-D Areas
Plane
Rectangle: A = l · w Square: A = s² Triangle: A = ½ · b · h Parallelogram:A = b · h Trapezoid: A = ½(b₁ + b₂) · h Circle: A = π r² Sector: A = ½ r² θ (θ in radians)
Perimeter & Circumference
Plane
Rectangle: P = 2(l + w) Square: P = 4s Triangle: P = a + b + c Circle: C = 2πr = πd Arc length: s = rθ (θ in radians)
Pythagorean Theorem
Core
a² + b² = c²
c is the hypotenuse (side opposite the right angle).
Common triples
3-4-5 · 5-12-13 8-15-17 · 7-24-25
Special right triangles
45-45-90: legs a, a → hyp = a√2 30-60-90: short a → long = a√3, hyp = 2a
3-D Volumes
Solid
Rectangular prism: V = l · w · h Cube: V = s³ Cylinder: V = π r² h Cone: V = ⅓ π r² h Sphere: V = (4/3) π r³ Pyramid: V = ⅓ · B · h
B = area of the base.
Surface Areas
Solid
Rectangular prism: SA = 2(lw + lh + wh) Cube: SA = 6s² Cylinder: SA = 2πr² + 2πrh Cone: SA = πr² + πrl l = slant height Sphere: SA = 4πr²
Angles & Polygons
Plane
Sum of interior angles
S = (n − 2) · 180°
Each interior angle (regular)
(n − 2) · 180° θ = ───────────── n
Sum of exterior angles
Always 360° (any convex polygon)
n = number of sides.
Law of Sines & Cosines
Triangles
Law of sines
a b c ───── = ───── = ───── sin A sin B sin C
Law of cosines
c² = a² + b² − 2ab · cos C
Area from two sides + angle
A = ½ ab · sin C
Heron's formula
s = (a + b + c) / 2 A = √[s(s−a)(s−b)(s−c)]
Right-Triangle Trig
Trig
sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse tan θ = opposite / adjacent csc θ = 1 / sin θ sec θ = 1 / cos θ cot θ = 1 / tan θ
SOH-CAH-TOA | Angles in degrees: 30°, 45°, 60° worth memorizing.
∫
Calculus Limits, differentiation rules, common derivatives, integration techniques, and the Fundamental Theorem of Calculus.
Limits
Core
Definition of derivative
f(x+h) − f(x) f'(x) = lim ───────────── h→0 h
Standard trig limit
lim sin(x)/x = 1 (x → 0) lim (1−cos x)/x = 0 (x → 0)
L'Hôpital's rule (0/0 or ∞/∞)
lim f(x)/g(x) = lim f'(x)/g'(x)
Differentiation Rules
Derivatives
Power rule: d/dx[xⁿ] = nxⁿ⁻¹ Constant: d/dx[c] = 0 Constant mult: d/dx[cf] = c·f' Sum/Diff: (f ± g)' = f' ± g'
Product rule
(fg)' = f'g + fg'
Quotient rule
d f f'g − fg' ──[───] = ────────── dx g g²
Chain rule
d/dx[f(g(x))] = f'(g(x)) · g'(x)
Common Derivatives
Derivatives
d/dx[sin x] = cos x d/dx[cos x] = −sin x d/dx[tan x] = sec² x d/dx[csc x] = −csc x · cot x d/dx[sec x] = sec x · tan x d/dx[cot x] = −csc² x d/dx[eˣ] = eˣ d/dx[aˣ] = aˣ · ln a d/dx[ln x] = 1/x d/dx[logₐx] = 1/(x · ln a) d/dx[arcsin x] = 1/√(1−x²) d/dx[arctan x] = 1/(1+x²)
Integration Rules
Integrals
Power rule
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ −1)
Special cases
∫ 1/x dx = ln|x| + C ∫ eˣ dx = eˣ + C ∫ aˣ dx = aˣ/ln a + C
Trig integrals
∫ sin x dx = −cos x + C ∫ cos x dx = sin x + C ∫ sec² x dx = tan x + C
Integration by parts
∫ u dv = uv − ∫ v du
Fundamental Theorem of Calculus
Core
Part I — differentiation
d x ─── ∫ f(t) dt = f(x) dx a
Part II — evaluation
b ∫ f(x) dx = F(b) − F(a) a where F'(x) = f(x)
Links the two central operations of calculus: differentiation and integration are inverses.
Taylor & Maclaurin Series
Series
Taylor series about x = a
∞ f⁽ⁿ⁾(a) f(x) = Σ ──────── (x−a)ⁿ n=0 n!
Common Maclaurin series
eˣ = 1 + x + x²/2! + x³/3! + … sin x = x − x³/3! + x⁵/5! − … cos x = 1 − x²/2! + x⁴/4! − … 1 ───── = 1 + x + x² + x³ + … |x|<1 1 − x
Critical Points & Optimization
Applications
Critical points
f'(x) = 0 or f'(x) undefined
Second derivative test
f''(c) > 0 → local minimum f''(c) < 0 → local maximum f''(c) = 0 → inconclusive
Concavity
f''(x) > 0 → concave up f''(x) < 0 → concave down inflection pt: f'' changes sign
Mean Value Theorem
f(b) − f(a) f'(c) = ───────────── for some c ∈ (a,b) b − a
Integration Techniques
Integrals
u-substitution
∫ f(g(x))·g'(x) dx = ∫ f(u) du where u = g(x), du = g'(x) dx
Integration by parts
∫ u dv = uv − ∫ v du Pick u using LIATE: Logs, Inverse trig, Algebraic, Trig, Exponential
Partial fractions
A(x) A B ──────────── = ───── + ───── (x+p)(x+q) x + p x + q