Commutative property of addition: a + b = b + a
Commutative property of multiplication: ab = ba
Associative property of addition: (a + b) + c = a + (b + c)
Associative property of multiplication: (ab)c = a(bc)
Distributive property: a(b + c) = ab + ac
Additive identity: a + 0 = a
Multiplicative identity: a × 1 = a
Additive inverse: a + (−a) = 0
Multiplicative inverse: a × (1/a) = 1, for a not zero
Zero property of multiplication: a × 0 = 0
Reflexive property: a = a
Symmetric property: If a = b, then b = a
Transitive property: If a = b and b = c, then a = c
Addition property of equality: If a = b, then a + c = b + c
Multiplication property of equality: If a = b, then ac = bc
Substitution property: If a = b, then b can replace a in any expression
Rule for multiplying two negatives: The product is positive
Rule for multiplying a negative and a positive: The product is negative
Rule for dividing two negatives: The quotient is positive
Subtracting a negative: a − (−b) = a + b
Solve x + 7 = 12: x = 5
Solve 3x = 21: x = 7
Solve 2x − 5 = 9: x = 7
Solve x/4 = 6: x = 24
Solve 5x + 3 = 2x + 15: x = 4
Rule for inequalities: Multiplying or dividing by a negative number flips the inequality sign
Solve −2x > 6: x < −3
Solve |x| = 5: x = 5 or x = −5
Solve |x| < 3: −3 < x < 3
Solve |x| > 3: x < −3 or x > 3
FOIL: First, Outer, Inner, Last: multiplying two binomials
Expand (x + 3)(x + 2): x² + 5x + 6
Expand (x + 4)²: x² + 8x + 16
Expand (x − 5)²: x² − 10x + 25
Expand (x + 3)(x − 3): x² − 9
Factor x² + 7x + 12: (x + 3)(x + 4)
Factor x² − 9: (x − 3)(x + 3)
Factor x² − 5x + 6: (x − 2)(x − 3)
Factor x² + 2x + 1: (x + 1)²
Factor 2x² + 6x: 2x(x + 3)
Sum of cubes: a³ + b³ = (a + b)(a² − ab + b²)
Difference of cubes: a³ − b³ = (a − b)(a² + ab + b²)
Binomial cube: (a + b)³ = a³ + 3a²b + 3ab² + b³
Solve x² = 49: x = 7 or x = −7
Solve x² − 5x + 6 = 0: x = 2 or x = 3
Midpoint of two numbers: Their average: (a + b)/2
Cross-multiplication: If a/b = c/d, then ad = bc
Adding fractions with unlike denominators: Use a common denominator first
Slope between two points: (y₂ − y₁)/(x₂ − x₁)
Perpendicular to slope 2: Slope −1/2
Distance, rate, time: d = r × t
Percent change: (new − old) / old × 100%
Simple interest: I = P r t
Average (mean) formula: Sum of values divided by number of values
Order of solving with variables on both sides: Collect variable terms on one side, constants on the other, then divide
Check a solution: Substitute it back into the original equation

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