The branch of mathematics dealing with symbols and the rules for manipulating those symbols. Algebra is the foundation for all higher mathematics.
What is Algebra?
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.
Variables and Expressions
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)
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:
Example: Solve 3x + 7 = 22
Step 1: Subtract 7 from both sides: 3x = 15
Step 2: Divide both sides by 3: x = 5
Check: 3(5) + 7 = 15 + 7 = 22 ✓
Slope-Intercept Form
y = mx + b
Where m is the slope (rate of change) and b is the y-intercept (where the line crosses the y-axis).
The slope m = (y₂ - y₁) / (x₂ - x₁) measures steepness
Positive slope: line goes up from left to right
Negative slope: line goes down from left to right
Zero slope: horizontal line
Undefined slope: vertical line
Inequalities
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.
Example
x² - 5x + 6 = 0
(x - 2)(x - 3) = 0
x = 2 or x = 3
2. Quadratic Formula
x = (-b ± √(b² - 4ac)) / (2a)
This formula works for any quadratic equation. The discriminant Δ = b² - 4ac determines the nature of the roots:
Δ > 0: Two distinct real roots
Δ = 0: One repeated real root
Δ < 0: Two complex conjugate roots
3. Completing the Square
Transform the equation into the form (x + p)² = q, then take the square root of both sides.
Example: Complete the square for x² + 6x + 2 = 0
x² + 6x = -2
x² + 6x + 9 = -2 + 9 (add (6/2)² = 9 to both sides)
(x + 3)² = 7
x + 3 = ±√7
x = -3 ± √7
The Vertex Form
y = a(x - h)² + k
The vertex of the parabola is at the point (h, k). If a > 0, the parabola opens upward; if a < 0, it opens downward.
Functions
A function is a rule that assigns to each input exactly one output. We write f(x) to denote the output of function f when the input is x.
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)
No solution: Lines are parallel (inconsistent)
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.
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.
Example: Solve x/(x − 2) + 3 = 2/(x − 2)
Step 1: Multiply both sides by (x − 2):
x + 3(x − 2) = 2
Step 2: Expand and solve: x + 3x − 6 = 2 → 4x = 8 → x = 2
Step 3: Check: x = 2 makes the denominator zero. No solution!
Logarithms
A logarithm answers the question: "What exponent do I need?" If bˣ = y, then log_b(y) = x. Logarithms are the inverse of exponential functions.
log_b(y) = x means bˣ = y
Common Logarithms
Common log: log(x) = log₁₀(x) — used in science and engineering
Natural log: ln(x) = logₑ(x) — used in calculus and advanced mathematics
Binary log: log₂(x) — used in computer science
Properties of Logarithms
Product rule: log_b(xy) = log_b(x) + log_b(y)
Quotient rule: log_b(x/y) = log_b(x) − log_b(y)
Power rule: log_b(xⁿ) = n · log_b(x)
Identity: log_b(b) = 1
Zero property: log_b(1) = 0
Change of base: log_b(x) = ln(x)/ln(b) = log(x)/log(b)
Example: Expand log₂(8x³/y)
= log₂(8x³) − log₂(y) (quotient rule)
= log₂(8) + log₂(x³) − log₂(y) (product rule)
= 3 + 3·log₂(x) − log₂(y) (since 2³ = 8 and power rule)
Solving Logarithmic Equations
Example: Solve log₃(2x + 1) = 4
Convert to exponential form: 3⁴ = 2x + 1
81 = 2x + 1
2x = 80
x = 40
Example: Solve ln(x) + ln(x − 2) = ln(3)
Step 1: Combine left side: ln(x(x − 2)) = ln(3)
Step 2: Since ln is one-to-one: x(x − 2) = 3
Step 3: Solve: x² − 2x − 3 = 0 → (x − 3)(x + 1) = 0 → x = 3 or x = −1
Step 4: Check domain: ln(x) requires x > 0, and ln(x − 2) requires x > 2.
So x = −1 is extraneous. Solution: x = 3
Solving Exponential Equations
When the variable is in the exponent, use logarithms to bring it down:
Example: Solve 5ˣ = 200
Take ln of both sides: x · ln(5) = ln(200)
x = ln(200)/ln(5) ≈ 5.298/1.609 ≈ 3.292
Example: Solve 3^(2x+1) = 7^(x−1)
Take ln of both sides: (2x + 1)ln(3) = (x − 1)ln(7)
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
Multiplication
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Use FOIL and remember that i² = −1.
Example: Multiply (3 + 2i)(1 − 4i)
= 3(1) + 3(−4i) + 2i(1) + 2i(−4i)
= 3 − 12i + 2i − 8i²
= 3 − 10i − 8(−1)
= 3 − 10i + 8 = 11 − 10i
Complex Conjugate
The conjugate of z = a + bi is z̄ = a − bi. Key property:
z · z̄ = a² + b² (always a non-negative real number)
Division
Multiply numerator and denominator by the conjugate of the denominator:
Example: Compute (2 + 3i)/(1 − i)
= (2 + 3i)(1 + i) / ((1 − i)(1 + i))
= (2 + 2i + 3i + 3i²) / (1 + 1)
= (2 + 5i − 3) / 2
= (−1 + 5i)/2 = −1/2 + 5i/2
The Complex Plane
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
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).
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