Knowing these formulas makes algebra much easier. They come up again and again in simplifying, factoring, and solving equations. This page collects the most important ones with examples for each.
1. Distributive Property
The distributive property lets you multiply a factor across terms inside parentheses:
Formula
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 a technique for multiplying two binomials:
Formula
(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
Formulas
(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
Formula
(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
Formulas
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
Used to find the roots of any quadratic equation ax² + bx + c = 0:
Formula
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
Formulas
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
These properties underlie every equation-solving step:
- 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
Formulas
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 |