📐 Complete K-12 Algebra Formulas 📐
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1. Basic Algebra Formulas
Properties of Operations
a + b = b + a
Commutative Property (Addition)
a × b = b × a
Commutative Property (Multiplication)
(a + b) + c = a + (b + c)
Associative Property (Addition)
(a × b) × c = a × (b × c)
Associative Property (Multiplication)
a(b + c) = ab + ac
Distributive Property
a + 0 = a
Identity Property (Addition)
a × 1 = a
Identity Property (Multiplication)
a × 0 = 0
Zero Property
Inverse Operations
a + (−a) = 0
Additive Inverse
a × 1a = 1 (a ≠ 0)
Multiplicative Inverse
Solving for Variable
If ax = b, then x = ba
If ax + b = c, then x = c − ba
2. Exponents & Radicals
Laws of Exponents
am × an = am+n
Product Rule
aman = am−n
Quotient Rule
(am)n = amn
Power Rule
(ab)n = anbn
Product to Power
(ab)n =
anbn
Quotient to Power
a0 = 1 (a ≠ 0)
Zero Exponent
a−n = 1an
Negative Exponent
am/n = nam = (na)m
Rational Exponent
Radical Rules
na × b =
na ×
nb
Product of Radicals
nab =
na
nb
Quotient of Radicals
mna =
mna
Nested Radicals
(na)n = a
Radical Power
Special Cases
a² = |a|
Square Root of Square
1n = 1
One to Any Power
3. Factoring Formulas
Difference of Squares
a² − b² = (a + b)(a − b)
Perfect Square Trinomials
a² + 2ab + b² = (a + b)²
a² − 2ab + b² = (a − b)²
Sum and Difference of Cubes
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
Cubic Expansions
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
Three Terms Square
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ac
Special Products
(a + b)(a − b) = a² − b²
(x + a)(x + b) = x² + (a + b)x + ab
(ax + b)(cx + d) = acx² + (ad + bc)x + bd
Higher Powers
a⁴ − b⁴ = (a² + b²)(a + b)(a − b)
a⁴ + a²b² + b⁴ = (a² + ab + b²)(a² − ab + b²)
Sophie Germain Identity
a⁴ + 4b⁴ = (a² + 2ab + 2b²)(a² − 2ab + 2b²)
4. Quadratic Formulas
Standard Form
ax² + bx + c = 0
Standard Quadratic Equation
Quadratic Formula
x =
−b ± b² − 4ac
2a
Discriminant
Δ = b² − 4ac
If Δ > 0: Two real solutions
If Δ = 0: One real solution (repeated root)
If Δ < 0: No real solutions (two complex solutions)
If Δ = 0: One real solution (repeated root)
If Δ < 0: No real solutions (two complex solutions)
Vertex Form
y = a(x − h)² + k
Vertex at (h, k)
Vertex Coordinates
h = −b2a
k = c − b²4a
Factored Form
y = a(x − r₁)(x − r₂)
r₁ and r₂ are roots
Sum and Product of Roots
r₁ + r₂ = −ba
Sum of Roots (Vieta's Formula)
r₁ × r₂ = ca
Product of Roots (Vieta's Formula)
Axis of Symmetry
x = −b2a
Completing the Square
x² + bx + (b2)² = (x + b2)²
5. Polynomial Formulas
General Form
P(x) = anxn + an-1xn-1 + ... + a₁x + a₀
Degree and Leading Coefficient
Degree: Highest power of x
Leading Coefficient: an (coefficient of highest power)
Leading Coefficient: an (coefficient of highest power)
Remainder Theorem
P(a) = Remainder when P(x) ÷ (x − a)
Factor Theorem
If P(a) = 0, then (x − a) is a factor of P(x)
Rational Root Theorem
Possible Rational Roots = ±
factors of constant term
factors of leading coefficient
Binomial Theorem
(a + b)n = Σ C(n,k) × an-k × bk
where C(n,k) = n!k!(n−k)!
Pascal's Triangle Expansions
(a + b)² = a² + 2ab + b²
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
Vieta's Formulas (General)
For polynomial anxn + ... + a₁x + a₀ = 0 with roots r₁, r₂, ..., rn:
Sum of roots = −an-1an
Product of roots = (−1)n × a₀an
Division Algorithm
P(x) = Q(x) × D(x) + R(x)
Polynomial = Quotient × Divisor + Remainder
6. Sequences & Series
Arithmetic Sequences
an = a₁ + (n − 1)d
nth term, where d = common difference
Sn = n2(a₁ + an)
Sum of n terms
Sn = n2[2a₁ + (n − 1)d]
Alternate sum formula
Geometric Sequences
an = a₁ × rn−1
nth term, where r = common ratio
Sn = a₁ × 1 − rn1 − r (r ≠ 1)
Sum of n terms
S∞ = a₁1 − r (|r| < 1)
Infinite sum (convergent)
Special Series
1 + 2 + 3 + ... + n = n(n + 1)2
Sum of first n natural numbers
1² + 2² + 3² + ... + n² = n(n + 1)(2n + 1)6
Sum of squares
1³ + 2³ + 3³ + ... + n³ = [n(n + 1)2]²
Sum of cubes
Arithmetic Mean
AM = a + b2
Geometric Mean
GM = ab
Harmonic Mean
HM =
2ab
a + b
Mean Relationship
AM ≥ GM ≥ HM
For positive numbers
7. Rational Expressions
Simplification
ac
bc
= ab (c ≠ 0)
Multiplication
ab ×
cd =
acbd
Division
ab ÷
cd =
ab ×
dc =
adbc
Addition (Same Denominator)
ac +
bc =
a + bc
Addition (Different Denominators)
ab +
cd =
ad + bcbd
Complex Fractions
ab
cd
=
ab ×
dc =
adbc
Negative Exponent in Fraction
(ab)−n =
(ba)n
8. Logarithmic Formulas
Basic Definition
logb(x) = y ⟺ by = x
Logarithm Laws
logb(xy) = logb(x) + logb(y)
Product Rule
logb(xy) = logb(x) − logb(y)
Quotient Rule
logb(xn) = n × logb(x)
Power Rule
Special Values
logb(1) = 0
logb(b) = 1
logb(bx) = x
blogb(x) = x
Change of Base Formula
logb(x) = loga(x)loga(b)
logb(x) = ln(x)ln(b)
Natural Logarithm (ln)
ln(x) = loge(x)
where e ≈ 2.71828
ln(e) = 1
ln(ex) = x
eln(x) = x
Common Logarithm
log(x) = log₁₀(x)
Exponential-Logarithm Relationship
ax = b ⟺ x = loga(b)
9. Coordinate Geometry Formulas
Distance Formula
d = (x₂ − x₁)² + (y₂ − y₁)²
Midpoint Formula
M = (x₁ + x₂2,
y₁ + y₂2)
Slope Formula
m = y₂ − y₁x₂ − x₁ =
riserun
Line Equations
y = mx + b
Slope-Intercept Form
y − y₁ = m(x − x₁)
Point-Slope Form
Ax + By = C
Standard Form
Parallel and Perpendicular Lines
Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = −1
Circle Equation
(x − h)² + (y − k)² = r²
Center: (h, k), Radius: r
Parabola (Vertex Form)
y = a(x − h)² + k
Vertical parabola, Vertex: (h, k)
x = a(y − k)² + h
Horizontal parabola, Vertex: (h, k)
Ellipse
(x − h)²a² +
(y − k)²b² = 1
Center: (h, k)
Hyperbola
(x − h)²a² −
(y − k)²b² = 1
Horizontal hyperbola, Center: (h, k)
10. Special Algebraic Formulas
Absolute Value Properties
|a × b| = |a| × |b|
|ab| =
|a||b|
|a + b| ≤ |a| + |b|
Triangle Inequality
Complex Numbers
i² = −1
(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
|a + bi| = a² + b²
Modulus (magnitude)
Matrix Operations
Matrix Addition: Add corresponding elements
Scalar Multiplication: Multiply each element by scalar
Matrix Multiplication: (AB)ij = Σ AikBkj
Scalar Multiplication: Multiply each element by scalar
Matrix Multiplication: (AB)ij = Σ AikBkj
Determinant (2×2)
det([a b; c d]) = ad − bc
Cramer's Rule
x = DxD,
y = DyD
For solving 2×2 systems
Inequality Properties
If a > b, then a + c > b + c
If a > b and c > 0, then ac > bc
If a > b and c < 0, then ac < bc
Function Composition
(f ∘ g)(x) = f(g(x))
Inverse Function
f(f−1(x)) = x
f−1(f(x)) = x
Direct Variation
y = kx
k = constant of variation
Inverse Variation
y = kx
Joint Variation
z = kxy
Compound Interest
A = P(1 + rn)nt
Compound interest formula
Exponential Growth/Decay
A = P(1 + r)t
Growth (r > 0) or Decay (0 < r < 1)
A = Pert
Continuous growth/decay
📋 Essential Formula Quick Reference
| Category | Key Formula | Name |
|---|---|---|
| Quadratic | x = (−b ± √(b²−4ac)) / 2a | Quadratic Formula |
| Distance | d = √((x₂−x₁)² + (y₂−y₁)²) | Distance Formula |
| Slope | m = (y₂−y₁)/(x₂−x₁) | Slope Formula |
| Factoring | a² − b² = (a+b)(a−b) | Difference of Squares |
| Exponent | am × an = am+n | Product Rule |
| Logarithm | log(xy) = log(x) + log(y) | Product Rule |
| Sequence | an = a₁ + (n−1)d | Arithmetic Sequence |
| Series | Sn = n(n+1)/2 | Sum of Naturals |
| Binomial | (a+b)² = a² + 2ab + b² | Perfect Square |
| Circle | (x−h)² + (y−k)² = r² | Circle Equation |
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