1. Distributive Property
The distributive property is what lets you multiply a term across everything inside a set of parentheses:
a(b + c) = ab + ac
a(b − c) = ab − ac
Examples:
3(x + 4) = 3x + 12
−2(5y − 7) = −10y + 14
a(b − c) = ab − ac
Examples:
3(x + 4) = 3x + 12
−2(5y − 7) = −10y + 14
2. FOIL Method
FOIL (First, Outer, Inner, Last) is just a memory aid for multiplying two binomials without missing a term:
(a + b)(c + d) = ac + ad + bc + bd
Example:
(x + 3)(x + 5)
= x·x + x·5 + 3·x + 3·5
= x² + 5x + 3x + 15
= x² + 8x + 15
Example:
(x + 3)(x + 5)
= x·x + x·5 + 3·x + 3·5
= x² + 5x + 3x + 15
= x² + 8x + 15
3. Perfect Square Trinomials
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
Examples:
(x + 4)² = x² + 8x + 16
(2y − 3)² = 4y² − 12y + 9
(a − b)² = a² − 2ab + b²
Examples:
(x + 4)² = x² + 8x + 16
(2y − 3)² = 4y² − 12y + 9
4. Difference of Two Squares
(a + b)(a − b) = a² − b²
Examples:
(x + 7)(x − 7) = x² − 49
(3a + 5)(3a − 5) = 9a² − 25
Examples:
(x + 7)(x − 7) = x² − 49
(3a + 5)(3a − 5) = 9a² − 25
5. Sum and Difference of Cubes
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
Examples:
x³ + 27 = (x + 3)(x² − 3x + 9)
8y³ − 1 = (2y − 1)(4y² + 2y + 1)
a³ − b³ = (a − b)(a² + ab + b²)
Examples:
x³ + 27 = (x + 3)(x² − 3x + 9)
8y³ − 1 = (2y − 1)(4y² + 2y + 1)
6. The Quadratic Formula
This is the formula to reach for when a quadratic ax² + bx + c = 0 won't factor cleanly:
x = (−b ± √(b² − 4ac)) / 2a
The expression b² − 4ac is the discriminant (Δ):
Δ > 0 → two distinct real roots
Δ = 0 → one repeated real root
Δ < 0 → no real roots (two complex roots)
Example: 2x² − 4x − 6 = 0
a=2, b=−4, c=−6
Δ = 16 + 48 = 64
x = (4 ± 8) / 4 → x = 3 or x = −1
The expression b² − 4ac is the discriminant (Δ):
Δ > 0 → two distinct real roots
Δ = 0 → one repeated real root
Δ < 0 → no real roots (two complex roots)
Example: 2x² − 4x − 6 = 0
a=2, b=−4, c=−6
Δ = 16 + 48 = 64
x = (4 ± 8) / 4 → x = 3 or x = −1
7. Exponent Rules
| Rule | Formula | Example |
|---|---|---|
| Product rule | aᵐ × aⁿ = aᵐ⁺ⁿ | x³ × x⁴ = x⁷ |
| Quotient rule | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | y⁵ ÷ y² = y³ |
| Power rule | (aᵐ)ⁿ = aᵐⁿ | (x²)³ = x⁶ |
| Zero exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| Negative exponent | a⁻ⁿ = 1/aⁿ | x⁻² = 1/x² |
| Fractional exponent | a^(m/n) = ⁿ√(aᵐ) | 8^(2/3) = (∛8)² = 4 |
| Product to power | (ab)ⁿ = aⁿbⁿ | (2x)³ = 8x³ |
8. Fraction Operations
Addition: a/b + c/d = (ad + bc) / bd
Subtraction: a/b − c/d = (ad − bc) / bd
Multiply: a/b × c/d = ac / bd
Divide: a/b ÷ c/d = a/b × d/c = ad / bc
Subtraction: a/b − c/d = (ad − bc) / bd
Multiply: a/b × c/d = ac / bd
Divide: a/b ÷ c/d = a/b × d/c = ad / bc
9. Properties of Equality
Every step you take when solving an equation is really one of these five properties in disguise:
- Addition property: If a = b, then a + c = b + c.
- Subtraction property: If a = b, then a − c = b − c.
- Multiplication property: If a = b, then ac = bc.
- Division property: If a = b and c ≠ 0, then a/c = b/c.
- Substitution: If a = b, you may replace a with b in any expression.
10. Slope and Linear Equations
Slope: m = (y₂ − y₁) / (x₂ − x₁)
Slope-intercept: y = mx + b
Point-slope form: y − y₁ = m(x − x₁)
Standard form: Ax + By = C
Slope-intercept: y = mx + b
Point-slope form: y − y₁ = m(x − x₁)
Standard form: Ax + By = C
Quick Reference Summary
| Formula Name | Formula |
|---|---|
| Distributive | a(b+c) = ab + ac |
| Perfect sq. (sum) | (a+b)² = a²+2ab+b² |
| Perfect sq. (diff) | (a−b)² = a²−2ab+b² |
| Diff. of squares | (a+b)(a−b) = a²−b² |
| Sum of cubes | a³+b³ = (a+b)(a²−ab+b²) |
| Diff. of cubes | a³−b³ = (a−b)(a²+ab+b²) |
| Quadratic formula | x = (−b±√(b²−4ac))/2a |