Algebra is the language of mathematical structure and symbolic reasoning. It turns patterns and quantitative relationships into expressions, equations, functions, and general rules that can be manipulated precisely.

1. Core ideas and notation

At its core, algebra replaces specific numbers with symbols so that one argument can describe many cases.

Variables, constants, and parameters

A variable represents a quantity that may change, such as $x$ or $t$.
A constant is a fixed value, such as $2$, $\pi$, or $-7$.
A parameter is a symbol treated as fixed within a problem, such as $m$ and $b$ in $y = mx + b$.

Equality

An equation states that two expressions have the same value:

$$ 2x + 3 = 11 $$

The goal is usually to find all values of the variable that make the statement true.

Sets of numbers

Set	Meaning	Examples
Natural numbers	Counting numbers	$1, 2, 3, \dots$
Integers	Whole numbers and negatives	$-3, 0, 8$
Rational numbers	Ratios of integers	$\frac{3}{4}, -2, 0.125$
Irrational numbers	Not expressible as ratios of integers	$\sqrt{2}, \pi$
Real numbers	All rational and irrational numbers on the line	most elementary algebra values

Order of operations

Evaluate in this order:

Parentheses and grouped expressions
Exponents and roots
Multiplication and division
Addition and subtraction

Multiplication distributes over addition:

$$ a(b + c) = ab + ac $$

This property is one of the most important tools in algebra.

2. Expressions and algebraic manipulation

An expression is a combination of numbers, variables, and operations, but without an equals sign.

Examples:

$$ 3x^2 - 5x + 2 $$

\frac{2a - 1}{a + 4}

Like terms

Only like terms can be combined.

$$ 4x + 7x = 11x $$

but

$$ 4x + 7x^2 $$

cannot be simplified by combining coefficients.

Distributing and factoring

Distributing expands an expression:

$$ 3(x - 4) = 3x - 12 $$

Factoring reverses the process:

$$ 3x - 12 = 3(x - 4) $$

Fractions in algebra

To add algebraic fractions, use a common denominator:

\frac{1}{x} + \frac{1}{x + 1} = \frac{x + 1 + x}{x(x + 1)} = \frac{2x + 1}{x(x + 1)}

Restrictions matter. In the example above:

x \ne 0, \quad x \ne -1

Identities

An identity is true for all allowed values of the variable.

Example:

$$ (a+b)^2 = a^2 + 2ab + b^2 $$

This is not an equation to solve; it is a pattern to use.

3. Linear equations and inequalities

A linear equation has variables only to the first power.

One-variable linear equations

Example:

$$ 5x - 7 = 18 $$

Solve by isolating the variable:

$$ 5x = 25 $$

$$ x = 5 $$

The same operation must be done to both sides to preserve equality.

Equations with variables on both sides

Example:

$$ 4x + 9 = 2x + 15 $$

Move variable terms together:

$$ 2x + 9 = 15 $$

$$ 2x = 6 $$

$$ x = 3 $$

Literal equations

Sometimes the goal is to solve for one variable in terms of others.

Example:

A = \frac{1}{2}bh

Solve for $h$:

$$ 2A = bh $$

h = \frac{2A}{b}

Inequalities

Solve inequalities much like equations, except for one crucial rule:

If you multiply or divide by a negative number, reverse the inequality sign.

Example:

$$ -2x > 8 $$

Divide by $-2$ and reverse the sign:

$$ x < -4 $$

Interval notation

Inequality	Interval
$x \ge 3$	$[3, \infty)$
$x < 2$	$(-\infty, 2)$
$-1 < x \le 4$	$(-1, 4]$

Linear equation explorer

Change slope and intercept to see how y = mx + b shifts and tilts on the coordinate plane.

Slope m 1.4 Intercept b -1

Equation y = 1.4x - 1.0

Intercepts x = 0.7, y = -1.0

4. Systems of equations

A system asks for values that satisfy multiple equations at once.

Two linear equations in two variables

General form:

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

Main methods

Method	Best use
Substitution	One variable is easy to isolate
Elimination	Coefficients line up well
Graphing	Visual interpretation is useful

Example: elimination

\begin{cases} 2x + y = 7 \\ 3x - y = 8 \end{cases}

Add the equations:

$$ 5x = 15 $$

$$ x = 3 $$

Substitute back:

2(3) + y = 7 \Rightarrow y = 1

So the solution is

$$ (x, y) = (3, 1) $$

Geometric meaning

One solution: the lines intersect once
No solution: the lines are parallel
Infinitely many solutions: the equations describe the same line

5. Exponents, radicals, and powers

Exponents compress repeated multiplication.

a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ factors}}

Exponent laws

For appropriate nonzero bases:

a^m a^n = a^{m+n}

\frac{a^m}{a^n} = a^{m-n}

(a^m)^n = a^{mn}

$$ (ab)^n = a^n b^n $$

a^{-n} = \frac{1}{a^n}

$$ a^0 = 1 $$

Rational exponents and radicals

a^{1/n} = \sqrt[n]{a}

a^{m/n} = \sqrt[n]{a^m}

For real-valued even roots, the radicand must be nonnegative:

\sqrt{x}

requires

x \ge 0

Simplifying radicals

\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}

Rationalizing a denominator

Example:

\frac{1}{\sqrt{3}} = \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3}

6. Polynomials and factoring

A polynomial is a sum of terms of the form $ax^n$, where $n$ is a nonnegative integer.

Examples:

Linear: $2x - 5$
Quadratic: $x^2 - 4x + 4$
Cubic: $x^3 + x$

Special products

These patterns appear constantly:

$$ (a+b)^2 = a^2 + 2ab + b^2 $$

$$ (a-b)^2 = a^2 - 2ab + b^2 $$

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

Common factoring methods

Method	Example
Greatest common factor	$6x^2 + 9x = 3x(2x+3)$
Difference of squares	$x^2 - 25 = (x-5)(x+5)$
Trinomial factoring	$x^2 + 5x + 6 = (x+2)(x+3)$
Grouping	$ax + ay + bx + by = (a+b)(x+y)$

Zero-product property

$$ ab = 0 $$

then

a = 0 \quad \text{or} \quad b = 0

This is the key step after factoring an equation.

Example:

$$ x^2 - 5x + 6 = 0 $$

$$ (x-2)(x-3) = 0 $$

x = 2 \quad \text{or} \quad x = 3

7. Rational expressions

A rational expression is a quotient of polynomials:

\frac{P(x)}{Q(x)}

where

Q(x) \ne 0

Simplifying

Factor first, then cancel common factors.

\frac{x^2 - 9}{x^2 - 3x} = \frac{(x-3)(x+3)}{x(x-3)} = \frac{x+3}{x}

Restrictions from the original denominator remain:

x \ne 0, \quad x \ne 3

Operations

Multiplication

\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}

Division

\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}

Addition and subtraction

Use a common denominator:

\frac{2}{x} - \frac{1}{x+1} = \frac{2(x+1) - x}{x(x+1)} = \frac{x+2}{x(x+1)}

Rational equations

Typical workflow:

State denominator restrictions.
Multiply both sides by the least common denominator.
Solve the resulting equation.
Check for extraneous values that make a denominator zero.

8. Quadratic equations and parabolas

A quadratic equation has the form

ax^2 + bx + c = 0, \quad a \ne 0

Main solution methods

Method	Best use
Factoring	Simple integer roots
Completing the square	Deriving structure or vertex form
Quadratic formula	General method

Completing the square

Start with

$$ x^2 + 6x - 7 = 0 $$

Move the constant:

$$ x^2 + 6x = 7 $$

Add $\left(\frac{6}{2}\right)^2 = 9$ to both sides:

$$ x^2 + 6x + 9 = 16 $$

$$ (x+3)^2 = 16 $$

x+3 = \pm 4

x = 1 \quad \text{or} \quad x = -7

Quadratic formula

For

$$ ax^2 + bx + c = 0 $$

the solutions are

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

The discriminant

\Delta = b^2 - 4ac

determines the root type:

Discriminant	Result
$\Delta > 0$	two distinct real roots
$\Delta = 0$	one repeated real root
$\Delta < 0$	two complex roots

Graph interpretation

The graph of a quadratic function is a parabola.

Vertex form:

$$ y = a(x-h)^2 + k $$

The vertex is

$$ (h, k) $$

If $a > 0$, the parabola opens upward. If $a < 0$, it opens downward.

9. Functions

A function assigns each input exactly one output.

Notation:

$$ f(x) $$

is read "f of x."

Domain and range

The domain is the set of allowed inputs.
The range is the set of outputs produced.

Example:

f(x) = \frac{1}{x-2}

Domain:

x \ne 2

Evaluating functions

$$ f(x) = 3x^2 - 1 $$

then

$$ f(2) = 3(2)^2 - 1 = 11 $$

Common function types

Type	Example	Key feature
Linear	$f(x)=mx+b$	constant rate of change
Quadratic	$f(x)=ax^2+bx+c$	parabolic graph
Rational	$f(x)=\frac{1}{x}$	asymptotes, restrictions
Exponential	$f(x)=ab^x$	multiplicative growth or decay
Absolute value	$f(x)=	x	$	piecewise V-shape

Function composition

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

Example:

f(x) = x+1, \quad g(x) = x^2

Then

(f \circ g)(x) = x^2 + 1

Inverse idea

An inverse undoes a function when it exists:

f(f^{-1}(x)) = x

Not every function has an inverse on its full domain.

10. Exponential and logarithmic relationships

Exponential functions model repeated multiplication:

$$ f(x) = ab^x $$

where $a \ne 0$, $b > 0$, and $b \ne 1$.

Growth and decay

If $b > 1$, the function grows.
If $0 < b < 1$, the function decays.

Logarithms

The logarithm is the inverse of exponentiation:

y = \log_b x \quad \Longleftrightarrow \quad b^y = x

Domain restriction:

$$ x > 0 $$

Log laws

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

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

\log_b(x^r) = r \log_b x

Solving an exponential equation

Example:

2^{x+1} = 16

Since

$$ 16 = 2^4 $$

we get

$$ x+1 = 4 $$

$$ x = 3 $$

Solving a logarithmic equation

Example:

\log_3(x-1) = 2

Convert to exponential form:

$$ x - 1 = 3^2 = 9 $$

$$ x = 10 $$

Check the domain:

$$ x - 1 > 0 $$

so the solution is valid.

11. Problem-solving workflow

When solving an algebra problem, use a disciplined sequence.

General workflow

Identify the object: expression, equation, inequality, system, or function.
Note all restrictions on allowed values.
Simplify structure before solving: distribute, factor, combine like terms, or clear denominators.
Choose a method that matches the form.
Solve carefully and keep operations reversible when possible.
Check the result in the original statement, especially for rational, radical, or logarithmic problems.

Choosing a method quickly

Form you see	Likely method
Linear equation	isolate the variable
Quadratic set equal to zero	factor or use quadratic formula
Denominators with variables	multiply by LCD after stating restrictions
Square root present	isolate the radical, then square both sides
Two linear equations	substitution or elimination
Exponential equation	rewrite with common base or use logs

Example workflow: rational equation

Solve

\frac{1}{x} + \frac{1}{2} = \frac{3}{4}

Restrictions:

x \ne 0

Multiply both sides by $4x$:

$$ 4 + 2x = 3x $$

$$ x = 4 $$

Check:

\frac{1}{4} + \frac{1}{2} = \frac{3}{4}

Valid.

12. Common pitfalls

Combining unlike terms, such as treating $x + x^2$ as $2x^2$.
Canceling terms across addition, such as
$\frac{x+3}{x} \ne 1+3$
Forgetting denominator restrictions in rational expressions.
Losing the $\pm$ when taking square roots:
$x^2 = 9 \Rightarrow x = \pm 3$
Failing to reverse an inequality after dividing by a negative number.
Squaring both sides and keeping an extraneous solution without checking.
Applying log laws to sums incorrectly:
$\log(x+y) \ne \log x + \log y$

13. Formula summary

Core identities

$$ a(b+c) = ab + ac $$

$$ (a+b)^2 = a^2 + 2ab + b^2 $$

$$ (a-b)^2 = a^2 - 2ab + b^2 $$

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

Exponent rules

a^m a^n = a^{m+n}

\frac{a^m}{a^n} = a^{m-n}

(a^m)^n = a^{mn}

a^{-n} = \frac{1}{a^n}

a^{m/n} = \sqrt[n]{a^m}

Linear and quadratic forms

$$ y = mx + b $$

$$ ax^2 + bx + c = 0 $$

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

Logarithms

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

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

\log_b(x^r) = r\log_b x

Sources

OpenStax Mathematics
Mathematics LibreTexts
Stewart, Calculus: Early Transcendentals
Lay, Linear Algebra and Its Applications
Rosen, Discrete Mathematics and Its Applications
Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems
Blitzstein and Hwang, Introduction to Probability
Parell GitHub repository