🧮 What is Algebra?
Algebra is a branch of mathematics that uses symbols (usually letters) to represent numbers and relationships. It allows us to create formulas, solve equations, and model real-world problems.
From solving for unknowns to manipulating expressions, Algebra is foundational in all areas of math, science, and data.
🔤 Variables, Constants, and Expressions
- Variable: A symbol (like x or y) that represents a number.
- Constant: A fixed value (like 2, -5, 0.75).
- Expression: A combination of variables, constants, and operations (e.g., 2x + 5).
Note: Expressions do not have equals signs, while equations do.
📏 Linear Equations
These are equations where the highest power of the variable is 1.
Standard form: ax + b = 0
Example: Solve 2x + 3 = 7 → x = 2
These equations graph as straight lines.
🔺 Quadratic Equations
Quadratic equations involve the square of a variable.
Standard form: ax² + bx + c = 0
📌 Solving Methods
- Factoring: Break into two binomials
- Quadratic Formula:
x = [-b ± √(b² - 4ac)] / 2a - Completing the Square
Example: x² - 5x + 6 = 0 → Factors to (x - 2)(x - 3) → x = 2 or 3
🔣 Polynomials
A polynomial is an expression with one or more terms that include variables raised to whole-number powers.
Example: 3x² + 2x - 5
Degree: Highest power of the variable
Leading coefficient: Coefficient of the highest-degree term
✂️ Factoring
Factoring rewrites polynomials as products of simpler expressions.
📌 Common Techniques:
- Common Factor: Factor out shared values (e.g., 2x + 4 → 2(x + 2))
- Difference of Squares: a² - b² = (a - b)(a + b)
- Trinomials: x² + bx + c → (x + m)(x + n)
⚖️ Inequalities
Inequalities are like equations but use <, ≤, >, ≥ instead of =.
Example: 2x - 3 < 7 → x < 5
Remember: When multiplying/dividing by a negative number, flip the inequality sign.
📈 Functions & Graphs
A function is a rule that assigns exactly one output to each input.
Notation: f(x) = 2x + 1
Graphs show how inputs and outputs change. Linear functions produce lines; quadratics produce parabolas.
🔁 Systems of Equations
Two or more equations with the same variables. Solve to find the point(s) where all are true.
Methods:
- Substitution: Solve one equation for a variable, plug into the other.
- Elimination: Add/subtract to eliminate a variable.
- Graphing: Graph both and find the intersection point.
🧱 Exponents and Radicals
Exponents:
aⁿ × aᵐ = aⁿ⁺ᵐ(aⁿ)ᵐ = aⁿ·ᵐa⁰ = 1a⁻ⁿ = 1/aⁿ
Radicals:
√ameans “what number squared equals a?”√(a²) = |a|- Radicals can be simplified and rationalized.
📉 Intro to Logarithms
Logarithms are the inverses of exponents.
logb(a) = c means bᶜ = a
Useful for solving equations with exponents as variables.
Example: log₂(8) = 3 because 2³ = 8