Algebra is one of the broadest and most fundamental branches of mathematics. At its core, algebra is about finding unknown values by using letters (called variables) to represent numbers in equations and formulas. The word "algebra" comes from the Arabic word al-jabr, meaning "reunion of broken parts," from the title of a 9th-century book by mathematician al-Khwarizmi.
Algebra provides the language and tools that are essential to nearly every area of mathematics, science, engineering, economics, and computer science. When you learn algebra, you're not just learning to solve equations — you're learning to think logically and abstractly.
Algebra is often called the "gatekeeper" subject because success in algebra opens the door to higher math courses like geometry, trigonometry, calculus, and beyond.
A variable is a symbol (usually a letter like x, y, or z) that represents an unknown or changeable value. An algebraic expression is a combination of variables, numbers, and operations.
Key Terminology
Constant: A fixed value, such as 5, -3, or π
Variable: A symbol representing an unknown value, like x or y
Coefficient: The number multiplied by a variable, e.g., in 7x, the coefficient is 7
Term: A single number, variable, or product of numbers and variables (e.g., 3x², -5y, 12)
Expression: A combination of terms connected by + or - signs (e.g., 3x² + 2x - 5)
Example: Simplifying Expressions
Simplify: 3x + 5y - 2x + 8y
Group like terms: (3x - 2x) + (5y + 8y) = x + 13y
Order of Operations (PEMDAS)
When evaluating expressions, follow this order:
Parentheses — evaluate expressions inside parentheses first
Exponents — evaluate powers and roots
Multiplication and Division — from left to right
Addition and Subtraction — from left to right
Linear Equations
A linear equation is an equation where the highest power of the variable is 1. The graph of a linear equation is always a straight line.
Standard Form
ax + b = c
Where a, b, and c are constants, and a ≠ 0.
Solving Linear Equations
The goal is to isolate the variable on one side of the equation using inverse operations:
An inequality compares two expressions using symbols like <, >, ≤, or ≥. Unlike equations, inequalities have a range of solutions.
Solving Inequalities
Solve inequalities the same way as equations, with one critical rule:
When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality sign.
Example: Solve -2x + 3 > 11
Step 1: Subtract 3 from both sides: -2x > 8
Step 2: Divide by -2 (flip the sign!): x < -4
The solution is all values of x less than -4.
Compound Inequalities
A compound inequality combines two inequalities joined by "and" or "or":
"And" inequality: Both conditions must be true (intersection). Example: -3 < x < 5
"Or" inequality: At least one condition must be true (union). Example: x < -2 or x > 4
Polynomials
A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents.
Classification by Degree
Constant: degree 0 (e.g., 7)
Linear: degree 1 (e.g., 3x + 2)
Quadratic: degree 2 (e.g., x² - 4x + 3)
Cubic: degree 3 (e.g., 2x³ + x² - 5x + 1)
Quartic: degree 4
Quintic: degree 5
Polynomial Operations
Addition/Subtraction: Combine like terms (same variable and exponent).
Multiplication: Use the distributive property (FOIL for binomials).
Example: FOIL Method
Multiply: (x + 3)(x - 5)
First: x · x = x²
Outer: x · (-5) = -5x
Inner: 3 · x = 3x
Last: 3 · (-5) = -15
Result: x² - 5x + 3x - 15 = x² - 2x - 15
Quadratic Equations
A quadratic equation has the standard form:
ax² + bx + c = 0, where a ≠ 0
Methods for Solving Quadratics
1. Factoring
If the quadratic can be written as a product of two binomials, set each factor equal to zero.
A system of equations is a set of two or more equations with the same variables. The solution is the set of values that satisfies all equations simultaneously.
Methods of Solving
1. Substitution Method
Solve one equation for one variable, then substitute into the other equation.
Example
Solve: y = 2x + 1 and 3x + y = 11
Substitute y = 2x + 1 into the second equation:
3x + (2x + 1) = 11
5x + 1 = 11
5x = 10, so x = 2
Then y = 2(2) + 1 = 5
Solution: (2, 5)
2. Elimination Method
Add or subtract equations to eliminate one variable.
3. Graphing Method
Graph both equations and find the intersection point(s).
Types of Solutions
One solution: Lines intersect at exactly one point (consistent and independent)
Infinitely many solutions: Lines are the same (consistent and dependent)
Exponents and Radicals
Exponents represent repeated multiplication. Understanding the laws of exponents is crucial for simplifying expressions.
Laws of Exponents
Product Rule: aᵐ · aⁿ = aᵐ⁺ⁿ
Quotient Rule: aᵐ / aⁿ = aᵐ⁻ⁿ
Power Rule: (aᵐ)ⁿ = aᵐⁿ
Zero Exponent: a⁰ = 1 (when a ≠ 0)
Negative Exponent: a⁻ⁿ = 1/aⁿ
Fractional Exponent: a^(m/n) = ⁿ√(aᵐ)
Radicals
A radical is the inverse operation of an exponent. The most common is the square root:
√a = a^(1/2)
Simplifying Radicals
Example: Simplify √72
√72 = √(36 × 2) = √36 × √2 = 6√2
Factoring Techniques
Factoring is the process of writing an expression as a product of simpler expressions. It's essential for solving equations and simplifying rational expressions.
Common Factoring Techniques
1. Greatest Common Factor (GCF)
Factor out the largest factor common to all terms:
6x³ + 9x² = 3x²(2x + 3)
2. Difference of Squares
a² - b² = (a + b)(a - b)
3. Perfect Square Trinomials
a² + 2ab + b² = (a + b)² a² - 2ab + b² = (a - b)²
4. Sum and Difference of Cubes
a³ + b³ = (a + b)(a² - ab + b²) a³ - b³ = (a - b)(a² + ab + b²)
5. Factoring by Grouping
Example: Factor x³ + 3x² + 2x + 6
Group: (x³ + 3x²) + (2x + 6)
Factor each group: x²(x + 3) + 2(x + 3)
Factor out (x + 3): (x + 3)(x² + 2)
Factoring is one of the most important skills in algebra. Practice recognizing patterns — over time, you'll be able to factor expressions quickly by sight.
Rational Expressions
A rational expression is a fraction whose numerator and denominator are both polynomials. Understanding rational expressions is essential for advanced algebra and calculus.
Simplifying Rational Expressions
Factor the numerator and denominator, then cancel common factors:
Always state the restrictions on the variable — values that make the original denominator zero are excluded from the domain, even after simplification.
Operations with Rational Expressions
Multiplication
(a/b) · (c/d) = ac/(bd)
Factor first, cancel common factors, then multiply what remains.
Division
(a/b) ÷ (c/d) = (a/b) · (d/c) = ad/(bc)
Multiply by the reciprocal of the divisor.
Addition and Subtraction
To add or subtract rational expressions, find a common denominator:
To solve an equation containing rational expressions, multiply both sides by the LCD to eliminate fractions, then solve the resulting polynomial equation. Always check for extraneous solutions — values that make the original denominators zero.
When we encounter equations like x² + 1 = 0, there is no real number solution because no real number squared gives −1. To solve such equations, mathematicians introduced the imaginary unit:
i = √(−1), so i² = −1
A complex number has the form a + bi, where a is the real part and b is the imaginary part.
Operations with Complex Numbers
Addition and Subtraction
(a + bi) + (c + di) = (a + c) + (b + d)i (a + bi) − (c + di) = (a − c) + (b − d)i
Complex numbers can be plotted on the complex plane (Argand diagram), where the horizontal axis represents the real part and the vertical axis represents the imaginary part.
Modulus (Absolute Value)
|a + bi| = √(a² + b²)
This is the distance from the origin to the point (a, b).
Polar Form
z = r(cos θ + i sin θ) = r · e^(iθ)
Where r = |z| is the modulus and θ = arg(z) is the argument (angle).
Euler's Formula: e^(iθ) = cos θ + i sin θ. Setting θ = π gives the famous Euler's identity: e^(iπ) + 1 = 0, linking five of the most important constants in mathematics.
Powers of i
The powers of i cycle with period 4:
i¹ = i
i² = −1
i³ = −i
i⁴ = 1
i⁵ = i (cycle repeats)
To find iⁿ, divide n by 4 and use the remainder: iⁿ = i^(n mod 4).
Sequences and Series
A sequence is an ordered list of numbers following a pattern. A series is the sum of the terms of a sequence.
Arithmetic Sequences
Each term differs from the previous by a constant common difference d:
A train leaves Station A heading east at 60 mph. Two hours later, another train leaves Station A heading east at 90 mph. How long does it take the second train to catch up?
Key formula: distance = rate × time
Let t = time (hours) after the second train departs.
First train's distance: 60(t + 2) (it had a 2-hour head start)
Second train's distance: 90t
They meet when distances are equal: 60(t + 2) = 90t
60t + 120 = 90t → 120 = 30t → t = 4 hours
The key to word problems is practice. The more you translate English into algebra, the more natural it becomes. Always define your variables clearly and check your answer in the original context (not just in the equation).
Explore Algebra Lessons
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