📐 Complete Reference · Updated 2026
Algebra Formulas
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Every formula you need — searchable, copyable, organized by topic. From linear equations to logarithm laws. Bookmark and come back any time.
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Linear Equations
8 formulasSlope-Intercept Form Basic
y = mx + b
m = slope, b = y-intercept. Most common form for graphing lines.
Point-Slope Form Basic
y − y₁ = m(x − x₁)
Use when you know a point (x₁, y₁) and slope m.
Standard Form Basic
Ax + By = C
A, B, C are integers, A ≥ 0. Used for systems of equations.
Slope Formula Basic
m = (y₂ − y₁) / (x₂ − x₁)
Rise over run. Slope between two points (x₁,y₁) and (x₂,y₂).
Midpoint Formula Basic
M = ((x₁+x₂)/2, (y₁+y₂)/2)
Midpoint between two coordinate points.
Distance Formula Intermediate
d = √((x₂−x₁)² + (y₂−y₁)²)
Straight-line distance between two coordinate points.
Parallel & Perpendicular Intermediate
Parallel: m₁ = m₂
Perpendicular: m₁ · m₂ = −1
Perpendicular: m₁ · m₂ = −1
Parallel lines share the same slope. Perpendicular slopes are negative reciprocals.
Absolute Value Equation Intermediate
|ax + b| = c → ax+b = c or ax+b = −c
Split into two equations. Valid only if c ≥ 0.
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Quadratic Equations
6 formulasQuadratic Formula Intermediate
x = (−b ± √(b²−4ac)) / 2a
Solves ax²+bx+c=0 for any values. The universal quadratic solver.
Discriminant Intermediate
Δ = b² − 4ac
Δ > 0: two real roots · Δ = 0: one real root · Δ < 0: no real roots
Vertex Form Intermediate
y = a(x − h)² + k
Vertex is at (h, k). Sign of a determines if parabola opens up or down.
Axis of Symmetry Intermediate
x = −b / 2a
Vertical line through the vertex of a parabola y = ax² + bx + c.
Vieta’s Formulas Advanced
x₁+x₂ = −b/a
x₁·x₂ = c/a
x₁·x₂ = c/a
Sum and product of roots of ax²+bx+c=0. Useful for factoring without solving.
Completing the Square Intermediate
x² + bx = (x + b/2)² − (b/2)²
Add (b/2)² to both sides to create a perfect square trinomial.
Factoring Identities
9 formulasDifference of Squares Basic
a² − b² = (a+b)(a−b)
Any expression of form a²−b² factors instantly. e.g. x²−9 = (x+3)(x−3)
Perfect Square Trinomial (+) Basic
a² + 2ab + b² = (a+b)²
e.g. x² + 6x + 9 = (x+3)². Middle term = 2·√first·√last
Perfect Square Trinomial (−) Basic
a² − 2ab + b² = (a−b)²
e.g. x² − 10x + 25 = (x−5)²
Sum of Cubes Intermediate
a³ + b³ = (a+b)(a²−ab+b²)
e.g. x³ + 8 = (x+2)(x²−2x+4). Mnemonic: SOAP — Same Opposite Always Positive
Difference of Cubes Intermediate
a³ − b³ = (a−b)(a²+ab+b²)
e.g. x³ − 27 = (x−3)(x²+3x+9)
Binomial Cube Intermediate
(a+b)³ = a³+3a²b+3ab²+b³
(a−b)³ = a³−3a²b+3ab²−b³
(a−b)³ = a³−3a²b+3ab²−b³
Coefficients follow Pascal’s triangle row 3: 1, 3, 3, 1
GCF Factoring Basic
ka + kb + kc = k(a + b + c)
Always check for GCF first before any other method. e.g. 6x²+9x = 3x(2x+3)
Trinomial: AC Method Intermediate
ax² + bx + c:
Find p,q: p·q=ac, p+q=b
→ factor by grouping
Find p,q: p·q=ac, p+q=b
→ factor by grouping
For 2x²+7x+3: ac=6, find 6+1=7 → 2x²+6x+x+3 = 2x(x+3)+1(x+3) = (2x+1)(x+3)
Fourth Power Difference Advanced
a⁴ − b⁴ = (a²+b²)(a+b)(a−b)
Factor first as difference of squares, then factor again. e.g. x⁴−16 = (x²+4)(x+2)(x−2)
Exponent Rules
10 formulasProduct Rule Basic
aᵐ · aⁿ = aᵐ⁺ⁿ
Same base, multiply → add exponents. e.g. x³·x² = x⁵
Quotient Rule Basic
aᵐ / aⁿ = aᵐ⁻ⁿ
Same base, divide → subtract exponents. e.g. x⁵/x² = x³
Power Rule Basic
(aᵐ)ⁿ = aᵐⁿ
Power to a power → multiply exponents. e.g. (x²)³ = x⁶
Zero Exponent Basic
a⁰ = 1 (a ≠ 0)
Any non-zero base raised to 0 equals 1. e.g. 5⁰ = 1, (−3)⁰ = 1
Negative Exponent Basic
a⁻ⁿ = 1 / aⁿ
Flip to the other side of the fraction. e.g. x⁻³ = 1/x³
Fractional Exponent Intermediate
a^(m/n) = ⁿ√(aᵐ) = (ⁿ√a)ᵐ
Denominator = root index, numerator = power. e.g. 8^(2/3) = (∛8)² = 4
Product to a Power Basic
(ab)ⁿ = aⁿbⁿ
Distribute exponent to each factor. e.g. (2x)³ = 8x³
Quotient to a Power Basic
(a/b)ⁿ = aⁿ / bⁿ
Distribute exponent to numerator and denominator. e.g. (x/3)² = x²/9
Scientific Notation Basic
a × 10ⁿ, 1 ≤ |a| < 10
e.g. 6,200 = 6.2×10³ · 0.0045 = 4.5×10⁻³
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Radical Rules Intermediate
√(ab) = √a · √b
√(a/b) = √a / √b
ⁿ√(aᵐ) = a^(m/n)
√(a/b) = √a / √b
ⁿ√(aᵐ) = a^(m/n)
Product and quotient rules for radicals. Used to simplify expressions.
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Logarithm Laws
9 formulasLogarithm Definition Intermediate
log_b(x) = y ↔ bʸ = x
“Log base b of x equals y” means b to the y equals x. e.g. log₂(8) = 3 because 2³ = 8
Log Product Rule Intermediate
log_b(xy) = log_b(x) + log_b(y)
Log of a product = sum of logs (same base). e.g. log(6) = log(2) + log(3)
Log Quotient Rule Intermediate
log_b(x/y) = log_b(x) − log_b(y)
Log of a quotient = difference of logs. e.g. log(5/2) = log(5) − log(2)
Log Power Rule Intermediate
log_b(xⁿ) = n · log_b(x)
Exponent comes down as coefficient. e.g. log(x³) = 3·log(x)
Change of Base Intermediate
log_b(x) = log(x) / log(b) = ln(x) / ln(b)
Convert any log to base 10 or natural log for calculator use.
Natural Logarithm Intermediate
ln(x) = log_e(x)
ln(eˣ) = x
e^ln(x) = x
ln(eˣ) = x
e^ln(x) = x
e ≈ 2.71828. ln and eˣ are inverse functions — they cancel each other.
Special Log Values Intermediate
log_b(1) = 0
log_b(b) = 1
log_b(bⁿ) = n
log_b(b) = 1
log_b(bⁿ) = n
These three are always true for any valid base b. Memorize them.
Solving Exponential Eq. Intermediate
bˣ = c → x = log(c) / log(b)
Take log of both sides. e.g. 2ˣ = 10 → x = log(10)/log(2) ≈ 3.32
Log–Exp Inverse Advanced
b^(log_b(x)) = x
log_b(bˣ) = x
log_b(bˣ) = x
Log and exponential with same base cancel. Core identity for simplification.
Systems of Equations
5 formulasSubstitution Method Basic
Solve eq.1 for y, substitute into eq.2, solve for x, back-substitute
Best when one variable is already isolated or easy to isolate.
Elimination Method Basic
Multiply eqs to get opposite coefficients, add to eliminate a variable
Best when coefficients are easy to match. Also called addition method.
Cramer’s Rule (2×2) Advanced
x = Dₓ/D, y = D_y/D
D = ad−bc
D = ad−bc
D = determinant of coefficient matrix. Dₓ, D_y use constants in place of column.
Number of Solutions Intermediate
1 solution: lines intersect
0 solutions: parallel lines
∞ solutions: same line
0 solutions: parallel lines
∞ solutions: same line
Consistent independent → 1 solution. Inconsistent → 0. Dependent → infinite.
Augmented Matrix Advanced
[a b | e]
[c d | f]
[c d | f]
Write system as matrix, apply row operations (swap, multiply, add) to reach row-echelon form.
Sequences & Series
8 formulasArithmetic: nth Term Intermediate
aₙ = a₁ + (n−1)d
a₁ = first term, d = common difference, n = term number.
Arithmetic Series Sum Intermediate
Sₙ = n/2 · (a₁ + aₙ)
Sₙ = n/2 · (2a₁ + (n−1)d)
Sₙ = n/2 · (2a₁ + (n−1)d)
Sum of first n terms. Use first form when you know first and last terms.
Geometric: nth Term Intermediate
aₙ = a₁ · rⁿ⁻¹
r = common ratio (each term × r = next). e.g. 2, 6, 18, 54… r = 3
Geometric Series (Finite) Intermediate
Sₙ = a₁ · (1 − rⁿ) / (1 − r), r ≠ 1
Sum of first n terms of a geometric sequence. Only valid when r ≠ 1.
Infinite Geometric Series Advanced
S∞ = a₁ / (1 − r), |r| < 1
Only converges when |r| < 1. If |r| ≥ 1, sum is infinite (diverges).
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Finding d and r Basic
Arithmetic: d = aₙ₊₁ − aₙ
Geometric: r = aₙ₊₁ / aₙ
Geometric: r = aₙ₊₁ / aₙ
Subtract consecutive terms for arithmetic (constant difference), divide for geometric (constant ratio).
Sigma Notation Intermediate
Σᵢ₌₁ⁿ i = n(n+1)/2
Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6
Σᵢ₌₁ⁿ i² = n(n+1)(2n+1)/6
Closed-form for common summations. Used in calculus and discrete math.
Binomial Theorem Advanced
(a+b)ⁿ = Σₖ₌₀ⁿ C(n,k) · aⁿ⁻ᵏ · bᵏ
C(n,k) = n! / (k!(n−k)!) = binomial coefficient. Coefficients from Pascal’s triangle.
Inequalities
6 formulasCore Properties Basic
a < b → a+c < b+c
a 0 → ac < bc
a < b, c bc ⚠️
a 0 → ac < bc
a < b, c bc ⚠️
CRITICAL: Multiplying or dividing by a negative number FLIPS the inequality sign.
Compound Inequalities Basic
AND: a < x < b → x ∈ (a,b)
OR: x b → x ∈ (−∞,a)∪(b,∞)
OR: x b → x ∈ (−∞,a)∪(b,∞)
AND = intersection (between). OR = union (outside). Write in interval notation.
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Absolute Value Ineq. Intermediate
|x| < a → −a < x < a
|x| > a → x a
|x| > a → x a
“Less than” → AND (between). “Greater than” → OR (outside). Valid for a > 0.
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Quadratic Inequality Intermediate
ax²+bx+c > 0:
Find roots, test intervals, check sign of a
Find roots, test intervals, check sign of a
Factor, find zeros, test each interval. If a>0: parabola opens up, positive outside roots.
Interval Notation Basic
(a,b): a < x < b (open)
[a,b]: a ≤ x ≤ b (closed)
[a,b): a ≤ x < b
[a,b]: a ≤ x ≤ b (closed)
[a,b): a ≤ x < b
Parenthesis = exclude endpoint. Bracket = include. ±∞ always uses parenthesis.
Rational Inequality Advanced
p(x)/q(x) > 0:
Find zeros of p and q, test intervals, exclude q=0
Find zeros of p and q, test intervals, exclude q=0
Critical values from both numerator AND denominator. Exclude denominator zeros from solution.
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How to Use This Algebra Formulas Reference
This cheat sheet covers every essential algebra formula from pre-algebra through Algebra II, organized into eight topic categories. Use the search bar at the top to find any formula instantly by keyword — type “quadratic”, “log”, “slope”, or any variable name. Use the topic tabs to filter by category. Click Copy on any formula card to copy it to your clipboard for use in notes or documents.
Each formula card shows the level tag (Basic / Intermediate / Advanced) so you can quickly identify which formulas are foundational and which are higher-level. The page is print-ready — press Ctrl+P (or Cmd+P on Mac) for a clean black-and-white printable version with no navigation or ads.
The Most Important Algebra Formulas to Know
If you’re studying for a test and need to prioritize, these are the formulas that appear most often in algebra coursework and exams:
Quadratic Formula
x = (−b ± √(b²−4ac)) / 2a is the single most important formula in all of Algebra I and II. It solves any quadratic equation ax² + bx + c = 0 regardless of whether it factors nicely. The discriminant b²−4ac tells you how many real solutions exist before you even solve: positive = two solutions, zero = one solution, negative = no real solutions.
Factoring Identities
The three special products — difference of squares (a²−b²), perfect square trinomials (a±b)², and sum/difference of cubes (a³±b³) — account for the majority of factoring problems in Algebra I through pre-calculus. Recognizing these patterns immediately saves significant time on exams.
Exponent Rules
The seven exponent rules (product, quotient, power, zero, negative, fractional, and product-to-a-power) are foundational for simplifying expressions and solving exponential equations. Pay particular attention to negative exponents (a⁻ⁿ = 1/aⁿ) and fractional exponents (a^(m/n) = ⁿ√aᵐ) — these cause the most confusion.
Logarithm Laws
The three core log laws (product rule, quotient rule, power rule) plus the change of base formula are everything you need for Algebra II logarithm problems. A common mistake is treating log(x + y) as log(x) + log(y) — this is wrong. The product rule applies to log(xy), not log(x+y).
Algebra Formulas by Grade Level
Pre-Algebra and Algebra I students need: slope formulas, the quadratic formula, basic exponent rules, GCF factoring, the difference of squares, and linear inequality properties. These appear in every state standard for grades 7–10.
Algebra II adds: logarithm laws, sequences and series formulas, Vieta’s formulas, rational inequalities, and the binomial theorem. If you’re preparing for SAT or ACT math, mastering the Algebra I formulas above is the highest return on study time — they appear in roughly 35% of math questions on both exams.