Algebra

1. Properties of Real Numbers & Expressions

•  Commutative: a + b = b + a    |   ab = ba

•  Associative: (a + b) + c = a + (b + c)    |   (ab)c = a(bc)

•  Distributive: a(b + c) = ab + ac    |   a(b − c) = ab − ac

•  Identity: a + 0 = a    |   a · 1 = a

•  Inverse: a + (−a) = 0    |   a · (1/a) = 1 (a ≠ 0)

•  Zero property: a · 0 = 0

Order of Operations: PEMDAS (Parentheses, Exponents, Multiply/Divide left→right, Add/Subtract left→right)

2. Exponents Rules

•  Product: a^m · a^n = a^(m+n)

•  Quotient: a^m / a^n = a^(m−n)    (a ≠ 0)

•  Power of power: (a^m)^n = a^(m·n)

•  Power of product: (ab)^n = a^n · b^n

•  Power of quotient: (a/b)^n = a^n / b^n

•  Zero exponent: a^0 = 1    (a ≠ 0)

•  Negative exponent: a^(−n) = 1 / a^n

•  Fractional: a^(m/n) = ⁿ√(a^m) = (ⁿ√a)^m

3. Linear Equations & Inequalities

•  Slope: m = (y₂ − y₁) / (x₂ − x₁)

•  Slope-intercept form: y = mx + b    (m = slope, b = y-intercept)

•  Point-slope form: y − y₁ = m(x − x₁)

•  Standard form: Ax + By = C

•  Horizontal line: y = k    (slope = 0)

•  Vertical line: x = k    (undefined slope)

•  Parallel lines: same slope

•  Perpendicular lines: slopes are negative reciprocals (m₁ · m₂ = −1)

Solving inequalities: Flip the inequality sign when multiplying/dividing by a negative number.

4. Systems of Linear Equations

•  Solution types: one solution (intersect), infinite (same line), none (parallel)

•  Substitution method

•  Elimination method (add/subtract equations to cancel a variable)

5. Quadratics

•  Standard form: ax² + bx + c = 0

•  Vertex form: y = a(x − h)² + k    → vertex at (h, k)

•  Factored form: y = a(x − r)(x − s)    → roots/zeros at r and s

•  Axis of symmetry: x = −b/(2a)    or x = h

•  Vertex: x = −b/(2a), then plug back in to find y

•  Quadratic formula: x = [−b ± √(b² − 4ac)] / (2a)

•  Discriminant D = b² − 4ac

•  D > 0 → 2 real roots

•  D = 0 → 1 real root (repeated)

•  D < 0 → no real roots (complex)

6. Factoring Patterns (most common)

•  Greatest Common Factor (GCF): pull out common term first

•  Difference of squares: a² − b² = (a − b)(a + b)

•  Perfect square trinomial:

•  a² + 2ab + b² = (a + b)²

•  a² − 2ab + b² = (a − b)²

•  Sum/difference of cubes:

•  x³ + a³ = (x + a)(x² − ax + a²)

•  x³ − a³ = (x − a)(x² + ax + a²)

•  Grouping: ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)

7. Distance, Midpoint, & Other Key Formulas

•  Distance between (x₁,y₁) and (x₂,y₂): d = √[(x₂−x₁)² + (y₂−y₁)²]

•  Midpoint: ((x₁+x₂)/2 , (y₁+y₂)/2)

•  Pythagorean theorem: a² + b² = c² (right triangle