Comprehensive Algebra Formulas Cheat Sheet

Basic Operations and Properties

Order of Operations (PEMDAS)

1. Parentheses (and other grouping symbols)
2. Exponents (and roots)
3. Multiplication and Division (from left to right)
4. Addition and Subtraction (from left to right)

Properties of Real Numbers

Exponents and Radicals

Laws of Exponents

Rules of Radicals

Polynomials

Basic Definitions

• Monomial: Expression with one term (e.g., $3x^2$)
• Binomial: Expression with two terms (e.g., $x^2 + 2x$)
• Trinomial: Expression with three terms (e.g., $x^2 + 2x + 1$)
• Polynomial: Expression with multiple terms (e.g., $ax^n + bx^{n-1} + … + k$)

Special Products

Factoring Formulas

Polynomial Division

Long division process:

1. Arrange polynomials in descending order
2. Divide the first term of the dividend by the first term of the divisor
3. Multiply the divisor by the result from step 2
4. Subtract this product from the dividend
5. Repeat with the remainder as the new dividend until the remainder degree is less than the divisor degree

Linear Equations and Inequalities

Linear Equation (Standard Form)

$ax + b = 0$ where $a \neq 0$

Solution: $x = -\frac{b}{a}$

Linear Equation (Point-Slope Form)

$y – y_1 = m(x – x_1)$ where $m$ is the slope and $(x_1, y_1)$ is a point on the line

Linear Equation (Slope-Intercept Form)

$y = mx + b$ where $m$ is the slope and $b$ is the y-intercept

Linear Equation (Two-Point Form)

$\frac{y – y_1}{y_2 – y_1} = \frac{x – x_1}{x_2 – x_1}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line

Slope Formula

$m = \frac{y_2 – y_1}{x_2 – x_1}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are two points on the line

Linear Inequality

$ax + b < 0$ or $ax + b > 0$ or $ax + b \leq 0$ or $ax + b \geq 0$ where $a \neq 0$

Note: When multiplying or dividing by a negative number, reverse the inequality sign.

Systems of Linear Equations

Two Equations with Two Unknowns

\begin{align} a_1x + b_1y &= c_1\ a_2x + b_2y &= c_2 \end{align}

Solution by Substitution

1. Solve one equation for one variable
2. Substitute into the other equation
3. Solve for the remaining variable
4. Substitute back to find the other variable

Solution by Elimination

1. Multiply equations to make coefficients of one variable equal in magnitude
2. Add or subtract equations to eliminate one variable
3. Solve for the remaining variable
4. Substitute back to find the other variable

Solutions by Cramers Rule

\begin{align} x = \frac{D_x}{D}, y = \frac{D_y}{D} \end{align}

where: \begin{align} D = \begin{vmatrix} a_1 & b_1 \ a_2 & b_2 \end{vmatrix} = a_1b_2 – a_2b_1\ D_x = \begin{vmatrix} c_1 & b_1 \ c_2 & b_2 \end{vmatrix} = c_1b_2 – c_2b_1\ D_y = \begin{vmatrix} a_1 & c_1 \ a_2 & c_2 \end{vmatrix} = a_1c_2 – a_2c_1 \end{align}

Quadratic Equations and Functions

Standard Form

$ax^2 + bx + c = 0$ where $a \neq 0$

Quadratic Formula

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

Discriminant

$\Delta = b^2 – 4ac$

• If $\Delta > 0$: Two distinct real roots
• If $\Delta = 0$: One real root (repeated)
• If $\Delta < 0$: Two complex conjugate roots

Completing the Square

1. Rewrite as $ax^2 + bx = -c$
2. Divide by $a$: $x^2 + \frac{b}{a}x = -\frac{c}{a}$
3. Add $(\frac{b}{2a})^2$ to both sides: $x^2 + \frac{b}{a}x + (\frac{b}{2a})^2 = -\frac{c}{a} + (\frac{b}{2a})^2$
4. Left side is now $(x + \frac{b}{2a})^2$
5. Simplify right side: $(x + \frac{b}{2a})^2 = \frac{b^2 – 4ac}{4a^2}$
6. Square root both sides: $x + \frac{b}{2a} = \pm\frac{\sqrt{b^2 – 4ac}}{2a}$
7. Solve for $x$: $x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$

Vieta’s Formulas

If $r$ and $s$ are the roots of $ax^2 + bx + c = 0$, then:

• $r + s = -\frac{b}{a}$
• $r \times s = \frac{c}{a}$

Quadratic Function Properties

For $f(x) = ax^2 + bx + c$ where $a \neq 0$:

• Vertex: $(-\frac{b}{2a}, f(-\frac{b}{2a}))$
• Axis of symmetry: $x = -\frac{b}{2a}$
• y-intercept: $(0, c)$
• x-intercepts: Solutions to $ax^2 + bx + c = 0$

Rational Expressions

Basic Operations

• Addition/Subtraction: $\frac{a}{b} \pm \frac{c}{d} = \frac{ad \pm bc}{bd}$
• Multiplication: $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$
• Division: $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$

Simplifying Rational Expressions

1. Factor numerator and denominator completely
2. Cancel common factors
3. Simplify remaining expression

Complex Fractions

Simplify by:

1. Method 1: Multiply numerator and denominator by LCD of all denominators
2. Method 2: Simplify numerator and denominator separately, then divide

Radicals and Rational Exponents

Converting Between Radicals and Rational Exponents

• $\sqrt[n]{a} = a^{\frac{1}{n}}$
• $\sqrt[n]{a^m} = a^{\frac{m}{n}}$

Simplifying Radical Expressions

1. Express radicand as product of perfect powers and remaining factors
2. Use property: $\sqrt[n]{a^n \times b} = a \times \sqrt[n]{b}$

Rationalizing Denominators

• For $\frac{a}{\sqrt{b}}$: Multiply by $\frac{\sqrt{b}}{\sqrt{b}}$ to get $\frac{a\sqrt{b}}{b}$
• For $\frac{a}{\sqrt{b} + \sqrt{c}}$: Multiply by $\frac{\sqrt{b} – \sqrt{c}}{\sqrt{b} – \sqrt{c}}$ to get $\frac{a(\sqrt{b} – \sqrt{c})}{b – c}$

Functions

Function Notation

$f(x) = $ expression in terms of $x$

Domain and Range

• Domain: Set of all possible input values
• Range: Set of all possible output values

Common Functions

Function Transformations

For a function $y = f(x)$:

• Vertical shift: $f(x) + k$ shifts $f(x)$ up by $k$ units
• Horizontal shift: $f(x – h)$ shifts $f(x)$ right by $h$ units
• Vertical stretch/compression: $a \cdot f(x)$ stretches by factor of $|a|$ (compression if $|a| < 1$)
• Horizontal stretch/compression: $f(bx)$ compresses by factor of $|b|$ (stretch if $|b| < 1$)
• Reflection across x-axis: $-f(x)$
• Reflection across y-axis: $f(-x)$

Function Composition

$(f \circ g)(x) = f(g(x))$

Inverse Functions

If $f(x) = y$, then $f^{-1}(y) = x$

To find $f^{-1}(x)$:

1. Replace $f(x)$ with $y$
2. Interchange $x$ and $y$
3. Solve for $y$
4. Replace $y$ with $f^{-1}(x)$

Logarithms

Definition

If $a^y = x$, then $\log_a(x) = y$ where $a > 0, a \neq 1, x > 0$

Properties of Logarithms

Common Logarithms

$\log_{10}(x)$ is often written as $\log(x)$

Natural Logarithms

$\log_e(x)$ is written as $\ln(x)$ where $e \approx 2.71828$

Logarithmic Equations

Strategies:

1. Use logarithm properties to combine/separate terms
2. Convert to exponential form
3. Combine like logarithmic terms
4. Check solutions (logarithms of negative numbers or zero are undefined)

Sequences and Series

Arithmetic Sequence

Terms differ by a constant value (common difference)

• General term: $a_n = a_1 + (n-1)d$ where $d$ is the common difference
• Sum of first n terms: $S_n = \frac{n}{2}(a_1 + a_n) = \frac{n}{2}[2a_1 + (n-1)d]$

Geometric Sequence

Terms form a constant ratio (common ratio)

• General term: $a_n = a_1 \cdot r^{n-1}$ where $r$ is the common ratio
• Sum of first n terms: $S_n = \frac{a_1(1-r^n)}{1-r}$ for $r \neq 1$
• Sum of infinite geometric series: $S_{\infty} = \frac{a_1}{1-r}$ for $|r| < 1$

Binomial Theorem

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

where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient

Complex Numbers

Basic Definitions

• Imaginary unit: $i = \sqrt{-1}$ where $i^2 = -1$
• Complex number: $z = a + bi$ where $a, b$ are real numbers
• Real part: $\text{Re}(z) = a$
• Imaginary part: $\text{Im}(z) = b$
• Complex conjugate: $\overline{z} = a – bi$

Operations with Complex Numbers

• Addition: $(a + bi) + (c + di) = (a + c) + (b + d)i$
• Subtraction: $(a + bi) – (c + di) = (a – c) + (b – d)i$
• Multiplication: $(a + bi)(c + di) = (ac – bd) + (ad + bc)i$
• Division: $\frac{a + bi}{c + di} = \frac{(a + bi)(c – di)}{(c + di)(c – di)} = \frac{ac + bd}{c^2 + d^2} + \frac{bc – ad}{c^2 + d^2}i$

Properties of Complex Numbers

• Modulus: $|a + bi| = \sqrt{a^2 + b^2}$
• Multiplicative inverse: $\frac{1}{a + bi} = \frac{a – bi}{a^2 + b^2}$
• Polar form: $a + bi = r(\cos\theta + i\sin\theta) = re^{i\theta}$ where $r = |a + bi|$ and $\theta = \tan^{-1}(\frac{b}{a})$
• De Moivre’s formula: $(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$

Conic Sections

Circle

Equation: $(x – h)^2 + (y – k)^2 = r^2$

• Center: $(h, k)$
• Radius: $r$

Parabola

Vertex at $(h, k)$:

• Vertical: $y = a(x – h)^2 + k$
• Horizontal: $x = a(y – k)^2 + h$

Focus-directrix definition:

• Vertical axis, vertex at origin: $y = \frac{x^2}{4p}$ (focus at $(0, p)$, directrix $y = -p$)
• Horizontal axis, vertex at origin: $x = \frac{y^2}{4p}$ (focus at $(p, 0)$, directrix $x = -p$)

Ellipse

Equation: $\frac{(x – h)^2}{a^2} + \frac{(y – k)^2}{b^2} = 1$

• Center: $(h, k)$
• Semi-major axis: $a$ (if $a > b$)
• Semi-minor axis: $b$ (if $a > b$)
• Foci: $(h \pm c, k)$ for horizontal ellipse or $(h, k \pm c)$ for vertical ellipse, where $c^2 = |a^2 – b^2|$
• Eccentricity: $e = \frac{c}{a}$ where $0 < e < 1$

Hyperbola

Equation: $\frac{(x – h)^2}{a^2} – \frac{(y – k)^2}{b^2} = 1$ (horizontal) or $\frac{(y – k)^2}{a^2} – \frac{(x – h)^2}{b^2} = 1$ (vertical)

• Center: $(h, k)$
• Vertices: $(h \pm a, k)$ for horizontal or $(h, k \pm a)$ for vertical
• Foci: $(h \pm c, k)$ for horizontal or $(h, k \pm c)$ for vertical, where $c^2 = a^2 + b^2$
• Asymptotes: $y – k = \pm\frac{b}{a}(x – h)$ for horizontal or $y – k = \pm\frac{a}{b}(x – h)$ for vertical
• Eccentricity: $e = \frac{c}{a}$ where $e > 1$

Matrices and Determinants

Matrix Operations

For matrices $A = [a_{ij}]$ and $B = [b_{ij}]$:

• Addition: $(A + B){ij} = a{ij} + b_{ij}$
• Scalar multiplication: $(cA){ij} = c \cdot a{ij}$
• Matrix multiplication: $(AB){ij} = \sum_k a{ik} \cdot b_{kj}$

Matrix Properties

• Identity matrix: $I_n$ has 1’s on the diagonal and 0’s elsewhere
• Transpose: $(A^T){ij} = a{ji}$
• Inverse: $A \cdot A^{-1} = A^{-1} \cdot A = I$

Determinant of 2×2 Matrix

$\det\begin{pmatrix} a & b \ c & d \end{pmatrix} = ad – bc$

Determinant of 3×3 Matrix

$\det\begin{pmatrix} a & b & c \ d & e & f \ g & h & i \end{pmatrix} = a(ei – fh) – b(di – fg) + c(dh – eg)$

Cramer’s Rule

For a system of $n$ linear equations with $n$ unknowns: $x_i = \frac{\det(A_i)}{\det(A)}$ where $A$ is the coefficient matrix and $A_i$ is the matrix formed by replacing the $i$-th column of $A$ with the constant terms.

Common Formulas and Identities

Quadratic Formula

$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$ for $ax^2 + bx + c = 0$

Distance Formula

$d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}$

Midpoint Formula

$M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$

Distance from Point to Line

$d = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}$ for line $ax + by + c = 0$ and point $(x_0, y_0)$

Trigonometric Identities

• $\sin^2\theta + \cos^2\theta = 1$
• $\tan\theta = \frac{\sin\theta}{\cos\theta}$
• $\cot\theta = \frac{\cos\theta}{\sin\theta}$
• $\sec\theta = \frac{1}{\cos\theta}$
• $\csc\theta = \frac{1}{\sin\theta}$

Pythagorean Theorem

$a^2 + b^2 = c^2$ for a right triangle with sides $a$, $b$, and hypotenuse $c$

Area Formulas

• Triangle: $A = \frac{1}{2}bh$ or $A = \frac{1}{2}ab\sin C$
• Rectangle: $A = lw$
• Parallelogram: $A = bh$
• Trapezoid: $A = \frac{1}{2}h(a + b)$ where $a$ and $b$ are parallel sides
• Circle: $A = \pi r^2$

Volume Formulas

• Cube: $V = s^3$
• Rectangular prism: $V = lwh$
• Sphere: $V = \frac{4}{3}\pi r^3$
• Cylinder: $V = \pi r^2h$
• Cone: $V = \frac{1}{3}\pi r^2h$