30 formulas

Algebra 2 Formula Sheet

These are the formulas an Algebra 2 course uses: quadratics and polynomials, radicals and rational exponents, logarithms, exponential models, sequences and series, functions, and counting and probability.

Each formula shows what its letters stand for. Open any one for a worked example and a calculator, or print the sheet to study from.

Search every formula

Quadratics and polynomials

Quadratic Formula
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Finds the roots of any quadratic equation ax² + bx + c = 0.
$a$
Coefficient of x²
$b$
Coefficient of x
$c$
The constant value
Worked example and calculator
Discriminant
$\Delta = b^2 - 4ac$
Determines the nature of roots in a quadratic equation.
$a$
Coefficient of x²
$b$
Coefficient of x
$c$
The constant value
Worked example and calculator
Vertex Formula
$x = \frac{-b}{2a}$
Finds the x-coordinate of the vertex of a parabola ax² + bx + c.
$a$
Coefficient of x²
$b$
Coefficient of x
Worked example and calculator
Sum of Cubes
$a^3 + b^3 = (a + b)(a^2 - a b + b^2)$
Factoring pattern for a sum of two perfect cubes.
$a$
Cube root of the first cube
$b$
Cube root of the second cube
Worked example and calculator
Difference of Cubes
$a^3 - b^3 = (a - b)(a^2 + a b + b^2)$
Factoring pattern for a difference of two perfect cubes.
$a$
Cube root of the first cube
$b$
Cube root of the second cube
Worked example and calculator
Polynomial Remainder Theorem
$P(x) = (x - c)Q(x) + P(c)$
When a polynomial P(x) is divided by (x − c), the remainder is the constant P(c), the value of P at x = c.
$P$
Polynomial in x
$c$
Constant c in divisor (x − c)
$Q$
Quotient polynomial from division
Worked example and calculator
Polynomial Factor Theorem
$(x - c) \text{ is a factor of } P(x) \iff P(c) = 0$
A linear factor (x − c) divides P(x) exactly when c is a root: substituting x = c gives zero.
$P$
Polynomial in x
$c$
Number to test as a root
Worked example and calculator
Rational Root Theorem (statement)
$\dfrac{p}{q}\ \text{(lowest terms)} \implies p \mid a_{0},\quad q \mid a_{n}$
Every possible rational root p/q (in lowest terms) must have p dividing the constant term and q dividing the leading coefficient, a finite checklist of candidates.
$p$
Integer factor of the constant term a₀
$q$
Integer factor of the leading coefficient aₙ
$a_0$
Coefficient of x⁰
$a_n$
Coefficient of highest power of x
Worked example and calculator
Binomial Theorem
$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$
Expanding (a + b)ⁿ: coefficients are binomial coefficients; each term has total degree n.
$a$
First part of the sum
$b$
Second part of the sum
$n$
Nonnegative integer power
$k$
Summation index from 0 to n
Worked example and calculator

Radicals and rational exponents

Rational Exponent as a Radical
$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m \quad (a > 0 \text{ typical})$
A rational exponent m/n means take the nth root, then raise to the mth power (or the equivalent order when defined).
$a$
Nonnegative base for real outputs
$m$
Integer power on the radical
$n$
Degree of the root (n ≥ 2)
Worked example and calculator
Rationalizing a Monomial Radical Denominator
$\frac{1}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a} \quad (a > 0)$
Multiply numerator and denominator by √a so the denominator becomes the rational number a.
$a$
Positive number under the square root
Worked example and calculator

Logarithms

Logarithm Definition
$\log_b a = c \iff b^c = a$
Logarithm is the inverse of exponentiation.
$b$
The base of the logarithm
$a$
The number being logged
$c$
The logarithm value
Worked example and calculator
Logarithm Product Rule
$\log_b(xy) = \log_b x + \log_b y$
Log of a product equals the sum of the logs.
$b$
The base of the logarithm
$x$
First number in the product
$y$
Second number in the product
Worked example and calculator
Logarithm Quotient Rule
$\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y$
Log of a quotient equals the difference of the logs.
$b$
The base of the logarithm
$x$
Number being divided
$y$
Number to divide by
Worked example and calculator
Logarithm Power Rule
$\log_b(x^p) = p\log_b x \quad (x > 0,\; b > 0,\; b \neq 1)$
An exponent inside the log moves out as a multiplier, the log version of a power law.
$b$
Logarithm base
$x$
Positive input
$p$
Real exponent on x
Worked example and calculator
Change-of-Base Formula
$\log_b x = \frac{\log_c x}{\log_c b} \quad (b, x, c > 0,\; b \neq 1,\; c \neq 1)$
Rewrite any base-b logarithm using a calculator-friendly base c (often 10 or e).
$b$
Base of the logarithm to convert
$x$
Positive input to the logarithm
$c$
Common or natural log base for computation
Worked example and calculator

Exponential models

Exponential Growth & Decay
$A = a(1 + r)^t$
A quantity that grows (r > 0) or decays (r < 0) by a fixed percent each period. Enter r as a decimal.
$a$
Starting value at t = 0
$r$
Decimal rate; positive for growth, negative for decay
$t$
Number of periods elapsed
Worked example and calculator
Compound Interest
$A = P\left(1 + \frac{r}{n}\right)^{n t}$
Final balance when interest compounds n times per year. Enter r as a decimal (0.05 for 5%).
$P$
Starting amount of money
$r$
Interest rate as a decimal, e.g. 0.05 for 5%
$n$
Compounds per year (12 = monthly, 4 = quarterly)
$t$
Length of time in years (years)
Worked example and calculator
Continuous Compound Growth
$A = Pe^{rt}$
When interest (or growth) compounds continuously at rate r per unit time, amount A grows exponentially from principal P, distinct from discrete compounding or half-life decay models.
$A$
Value after time t
$P$
Starting amount
$r$
Nominal continuous rate per time unit
$t$
Elapsed time (time units)
Worked example and calculator

Sequences and series

Arithmetic Sequence (nth Term)
$a_n = a_1 + (n - 1)d$
Each term adds a common difference d to the previous one; aₙ is the value at position n.
$a_n$
Term at index n
$a_1$
Starting value
$n$
Term number (positive integer)
$d$
Added each step
Worked example and calculator
Arithmetic Series (Partial Sum)
$S_n = \frac{n}{2}(a_1 + a_n)$
Sum of the first n terms of an arithmetic sequence, average of first and last times how many terms.
$S_n$
Sum of first n terms
$n$
Count of terms summed
$a_1$
First term
$a_n$
Last term in the sum
Worked example and calculator
Geometric Sequence (nth Term)
$a_n = a_1 r^{n - 1} \quad (r \neq 0 \text{ typical})$
Each term multiplies the previous by common ratio r; growth or decay by a constant factor.
$a_n$
Term at index n
$a_1$
Starting value
$r$
Factor between successive terms
$n$
Term number
Worked example and calculator
Finite Geometric Series Sum
$S_n = a_1 \frac{1 - r^n}{1 - r} \quad (r \neq 1)$
Sum of the first n terms of a geometric sequence; for r = 1 use Sₙ = n·a₁ instead.
$S_n$
Sum of first n terms
$a_1$
First term
$r$
Ratio between terms (not equal to 1)
$n$
How many terms
Worked example and calculator
Infinite Geometric Series
$S = \frac{a}{1 - r}, \quad |r| < 1$
The sum of an infinite geometric series with first term a and common ratio r, valid only when |r| < 1.
$a$
The first term of the series
$r$
Each term times r gives the next; must satisfy |r| < 1
Worked example and calculator

Functions and complex numbers

Function Composition
$(f \circ g)(x) = f\bigl(g(x)\bigr)$
Applying one function to the output of another: do g first, then feed the result into f.
$f$
Applied second, to the output of g
$g$
Applied first, to the input x
$x$
The value fed into the inner function
Worked example and calculator
Inverse Function
$f\bigl(f^{-1}(x)\bigr) = x \quad\text{and}\quad f^{-1}\bigl(f(x)\bigr) = x$
An inverse function undoes the original: swap x and y and solve for y. Each composition returns the input.
$f$
The function to invert
$f^{-1}$
Undoes f
$x$
The value passed through both functions
Worked example and calculator
Modulus of a Complex Number
$|a + bi| = \sqrt{a^2 + b^2}$
The size (distance from the origin) of a complex number a + bi, where i is the imaginary unit with i² = −1.
$a$
The real component of a + bi
$b$
The coefficient of i
Worked example and calculator

Counting and probability

Permutations
${}_nP_r = \frac{n!}{(n - r)!}$
The number of ways to arrange r items from n distinct items when order matters.
$n$
How many distinct items are available
$r$
How many are arranged (0 ≤ r ≤ n)
Worked example and calculator
Combinations
${}_nC_r = \frac{n!}{r!\,(n - r)!}$
The number of ways to choose r items from n distinct items when order does NOT matter.
$n$
How many distinct items are available
$r$
How many are selected (0 ≤ r ≤ n)
Worked example and calculator
Basic Probability
$P(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$
The probability of an event with equally likely outcomes: favorable outcomes divided by total outcomes.
$f$
Number of outcomes that count as the event
$N$
Total number of equally likely outcomes
Worked example and calculator

Learn these formulas

Other formula sheets

Algebra 2 Formula Sheet

Each formula shows what its letters stand for. Open any one for a worked example and a calculator, or print the sheet to study from.

Quadratics and polynomials

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Finds the roots of any quadratic equation ax² + bx + c = 0.

$a$: Coefficient of x²
$b$: Coefficient of x
$c$: The constant value

Worked example and calculator

Discriminant

\Delta = b^2 - 4ac

Determines the nature of roots in a quadratic equation.

$a$: Coefficient of x²
$b$: Coefficient of x
$c$: The constant value

Worked example and calculator

Vertex Formula

x = \frac{-b}{2a}

Finds the x-coordinate of the vertex of a parabola ax² + bx + c.

$a$: Coefficient of x²
$b$: Coefficient of x

Worked example and calculator

Sum of Cubes

a^3 + b^3 = (a + b)(a^2 - a b + b^2)

Factoring pattern for a sum of two perfect cubes.

$a$: Cube root of the first cube
$b$: Cube root of the second cube

Worked example and calculator

Difference of Cubes

a^3 - b^3 = (a - b)(a^2 + a b + b^2)

Factoring pattern for a difference of two perfect cubes.

$a$: Cube root of the first cube
$b$: Cube root of the second cube

Worked example and calculator

Polynomial Remainder Theorem

P(x) = (x - c)Q(x) + P(c)

When a polynomial P(x) is divided by (x − c), the remainder is the constant P(c), the value of P at x = c.

$P$: Polynomial in x
$c$: Constant c in divisor (x − c)
$Q$: Quotient polynomial from division

Worked example and calculator

Polynomial Factor Theorem

(x - c) \text{ is a factor of } P(x) \iff P(c) = 0

A linear factor (x − c) divides P(x) exactly when c is a root: substituting x = c gives zero.

$P$: Polynomial in x
$c$: Number to test as a root

Worked example and calculator

Rational Root Theorem (statement)

\dfrac{p}{q}\ \text{(lowest terms)} \implies p \mid a_{0},\quad q \mid a_{n}

Every possible rational root p/q (in lowest terms) must have p dividing the constant term and q dividing the leading coefficient, a finite checklist of candidates.

$p$: Integer factor of the constant term a₀
$q$: Integer factor of the leading coefficient aₙ
$a_0$: Coefficient of x⁰
$a_n$: Coefficient of highest power of x

Worked example and calculator

Binomial Theorem

(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Expanding (a + b)ⁿ: coefficients are binomial coefficients; each term has total degree n.

$a$: First part of the sum
$b$: Second part of the sum
$n$: Nonnegative integer power
$k$: Summation index from 0 to n

Worked example and calculator

Radicals and rational exponents

Rational Exponent as a Radical

a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m \quad (a > 0 \text{ typical})

A rational exponent m/n means take the nth root, then raise to the mth power (or the equivalent order when defined).

$a$: Nonnegative base for real outputs
$m$: Integer power on the radical
$n$: Degree of the root (n ≥ 2)

Worked example and calculator

Rationalizing a Monomial Radical Denominator

\frac{1}{\sqrt{a}} \cdot \frac{\sqrt{a}}{\sqrt{a}} = \frac{\sqrt{a}}{a} \quad (a > 0)

Multiply numerator and denominator by √a so the denominator becomes the rational number a.

$a$: Positive number under the square root

Worked example and calculator

Logarithms

Logarithm Definition

\log_b a = c \iff b^c = a

Logarithm is the inverse of exponentiation.

$b$: The base of the logarithm
$a$: The number being logged
$c$: The logarithm value

Worked example and calculator

Logarithm Product Rule

\log_b(xy) = \log_b x + \log_b y

Log of a product equals the sum of the logs.

$b$: The base of the logarithm
$x$: First number in the product
$y$: Second number in the product

Worked example and calculator

Logarithm Quotient Rule

\log_b\left(\frac{x}{y}\right) = \log_b x - \log_b y

Log of a quotient equals the difference of the logs.

$b$: The base of the logarithm
$x$: Number being divided
$y$: Number to divide by

Worked example and calculator

Logarithm Power Rule

\log_b(x^p) = p\log_b x \quad (x > 0,\; b > 0,\; b \neq 1)

An exponent inside the log moves out as a multiplier, the log version of a power law.

$b$: Logarithm base
$x$: Positive input
$p$: Real exponent on x

Worked example and calculator

Change-of-Base Formula

\log_b x = \frac{\log_c x}{\log_c b} \quad (b, x, c > 0,\; b \neq 1,\; c \neq 1)

Rewrite any base-b logarithm using a calculator-friendly base c (often 10 or e).

$b$: Base of the logarithm to convert
$x$: Positive input to the logarithm
$c$: Common or natural log base for computation

Worked example and calculator

Exponential models

Exponential Growth & Decay

A = a(1 + r)^t

A quantity that grows (r > 0) or decays (r < 0) by a fixed percent each period. Enter r as a decimal.

$a$: Starting value at t = 0
$r$: Decimal rate; positive for growth, negative for decay
$t$: Number of periods elapsed

Worked example and calculator

Compound Interest

A = P\left(1 + \frac{r}{n}\right)^{n t}

Final balance when interest compounds n times per year. Enter r as a decimal (0.05 for 5%).

$P$: Starting amount of money
$r$: Interest rate as a decimal, e.g. 0.05 for 5%
$n$: Compounds per year (12 = monthly, 4 = quarterly)
$t$: Length of time in years (years)

Worked example and calculator

Continuous Compound Growth

A = Pe^{rt}

When interest (or growth) compounds continuously at rate r per unit time, amount A grows exponentially from principal P, distinct from discrete compounding or half-life decay models.

$A$: Value after time t
$P$: Starting amount
$r$: Nominal continuous rate per time unit
$t$: Elapsed time (time units)

Worked example and calculator

Sequences and series

Arithmetic Sequence (nth Term)

a_n = a_1 + (n - 1)d

Each term adds a common difference d to the previous one; aₙ is the value at position n.

$a_n$: Term at index n
$a_1$: Starting value
$n$: Term number (positive integer)
$d$: Added each step

Worked example and calculator

Arithmetic Series (Partial Sum)

S_n = \frac{n}{2}(a_1 + a_n)

Sum of the first n terms of an arithmetic sequence, average of first and last times how many terms.

$S_n$: Sum of first n terms
$n$: Count of terms summed
$a_1$: First term
$a_n$: Last term in the sum

Worked example and calculator

Geometric Sequence (nth Term)

a_n = a_1 r^{n - 1} \quad (r \neq 0 \text{ typical})

Each term multiplies the previous by common ratio r; growth or decay by a constant factor.

$a_n$: Term at index n
$a_1$: Starting value
$r$: Factor between successive terms
$n$: Term number

Worked example and calculator

Finite Geometric Series Sum

S_n = a_1 \frac{1 - r^n}{1 - r} \quad (r \neq 1)

Sum of the first n terms of a geometric sequence; for r = 1 use Sₙ = n·a₁ instead.

$S_n$: Sum of first n terms
$a_1$: First term
$r$: Ratio between terms (not equal to 1)
$n$: How many terms

Worked example and calculator

Infinite Geometric Series

S = \frac{a}{1 - r}, \quad |r| < 1

The sum of an infinite geometric series with first term a and common ratio r, valid only when |r| < 1.

$a$: The first term of the series
$r$: Each term times r gives the next; must satisfy |r| < 1

Worked example and calculator

Functions and complex numbers

Function Composition

(f \circ g)(x) = f\bigl(g(x)\bigr)

Applying one function to the output of another: do g first, then feed the result into f.

$f$: Applied second, to the output of g
$g$: Applied first, to the input x
$x$: The value fed into the inner function

Worked example and calculator

Inverse Function

f\bigl(f^{-1}(x)\bigr) = x \quad\text{and}\quad f^{-1}\bigl(f(x)\bigr) = x

An inverse function undoes the original: swap x and y and solve for y. Each composition returns the input.

$f$: The function to invert
$f^{-1}$: Undoes f
$x$: The value passed through both functions

Worked example and calculator

Modulus of a Complex Number

|a + bi| = \sqrt{a^2 + b^2}

The size (distance from the origin) of a complex number a + bi, where i is the imaginary unit with i² = −1.

$a$: The real component of a + bi
$b$: The coefficient of i

Worked example and calculator

Counting and probability

Permutations

{}_nP_r = \frac{n!}{(n - r)!}

The number of ways to arrange r items from n distinct items when order matters.

$n$: How many distinct items are available
$r$: How many are arranged (0 ≤ r ≤ n)

Worked example and calculator

Combinations

{}_nC_r = \frac{n!}{r!\,(n - r)!}

The number of ways to choose r items from n distinct items when order does NOT matter.

$n$: How many distinct items are available
$r$: How many are selected (0 ≤ r ≤ n)

Worked example and calculator

Basic Probability

P(E) = \frac{\text{favorable outcomes}}{\text{total outcomes}}

The probability of an event with equally likely outcomes: favorable outcomes divided by total outcomes.

$f$: Number of outcomes that count as the event
$N$: Total number of equally likely outcomes

Worked example and calculator