Algebra
Linear Equations
y = mx + b—Slope-intercept form
Ax + By = C—Standard form
m = (y₂-y₁)/(x₂-x₁)—Slope formula
y - y₁ = m(x - x₁)—Point-slope form
Parallel lines: same slope—m₁ = m₂
Perpendicular: m₁m₂ = -1—Negative reciprocal slopes
x-intercept: set y = 0—Where line crosses x-axis
y-intercept: set x = 0—Where line crosses y-axis
Quadratics
ax² + bx + c = 0—Standard quadratic form
x = (-b±√(b²-4ac))/2a—Quadratic formula
b² - 4ac > 0—Two real roots
b² - 4ac = 0—One repeated root
b² - 4ac < 0—No real roots (complex)
a(x-h)² + k—Vertex form, vertex (h,k)
a(x-r₁)(x-r₂)—Factored form with roots r₁, r₂
Vertex: x = -b/(2a)—x-coordinate of vertex
Sum of roots = -b/a—Vieta's formula
Product of roots = c/a—Vieta's formula
Exponent Rules
aᵐ × aⁿ = aᵐ⁺ⁿ—Product rule
aᵐ / aⁿ = aᵐ⁻ⁿ—Quotient rule
(aᵐ)ⁿ = aᵐⁿ—Power rule
a⁰ = 1—Zero exponent
a⁻ⁿ = 1/aⁿ—Negative exponent
(ab)ⁿ = aⁿbⁿ—Product to power
(a/b)ⁿ = aⁿ/bⁿ—Quotient to power
a^(1/n) = ⁿ√a—Fractional exponent
a^(m/n) = ⁿ√(aᵐ)—Rational exponent
Logarithms
log_b(x) = y ⟺ bʸ = x—Log definition
log(ab) = log a + log b—Product rule
log(a/b) = log a - log b—Quotient rule
log(aⁿ) = n log a—Power rule
ln(x) = log_e(x)—Natural logarithm
log_b(a) = ln a / ln b—Change of base
log_b(1) = 0—Log of 1
log_b(b) = 1—Log of base
Sequences & Series
aₙ = a₁ + (n-1)d—Arithmetic nth term
Sₙ = n(a₁ + aₙ)/2—Arithmetic series sum
aₙ = a₁ × rⁿ⁻¹—Geometric nth term
Sₙ = a₁(1-rⁿ)/(1-r)—Geometric finite sum
S∞ = a₁/(1-r), |r|<1—Infinite geometric sum
Σᵢ₌₁ⁿ i = n(n+1)/2—Sum of first n integers
Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6—Sum of squares
Factoring Patterns
a² - b² = (a+b)(a-b)—Difference of squares
a² + 2ab + b² = (a+b)²—Perfect square trinomial
a² - 2ab + b² = (a-b)²—Perfect square trinomial
a³ + b³ = (a+b)(a²-ab+b²)—Sum of cubes
a³ - b³ = (a-b)(a²+ab+b²)—Difference of cubes
ac method for ax²+bx+c—Find factors of ac that sum to b
Inequalities & Absolute Value
|x| = a → x = ±a—Absolute value equation
|x| < a → -a < x < a—Absolute value inequality
|x| > a → x<-a or x>a—Absolute value inequality
Flip sign when × or ÷ by negative—Inequality rule
Triangle ineq: |a+b| ≤ |a|+|b|—Triangle inequality
Systems of Equations
Substitution method—Solve one eq, plug into other
Elimination method—Add/subtract to cancel variable
Consistent: at least one solution—Lines intersect
Inconsistent: no solution—Parallel lines
Dependent: infinite solutions—Same line
Cramer's rule: x = Dₓ/D—Using determinants
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