Algebra Basics

Basic Concepts

• Algebra is a branch of mathematics that uses symbols (typically letters) to represent numbers and quantities in equations and formulas. It involves manipulating these symbols to solve for unknowns.
• Variables are symbols that represent unknown values. Constants are fixed numerical values.
• An equation is a statement that asserts the equality of two expressions. It contains an equals sign (=).
• Expressions are combinations of variables, constants, and mathematical operations (addition, subtraction, multiplication, division, etc.). They do not contain an equals sign.

Fundamental Operations

• Addition: Combining two or more quantities. The sum of two numbers a and b is represented as 'a + b'.
• Subtraction: Finding the difference between two quantities. The difference between two numbers a and b is 'a - b'.
• Multiplication: Repeated addition. The product of two numbers a and b is 'a * b' or 'a x b' or 'ab'.
• Division: Finding how many times one quantity is contained within another. The quotient of two numbers a and b is 'a / b' or 'a ÷ b'.
• Order of Operations (PEMDAS/BODMAS): A set of rules specifying the sequence in which calculations should be performed. Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right).

Algebraic Equations

• An algebraic equation states a relationship between variables and constants.
• Solving an equation means finding the value(s) of the variable(s) that make the equation true.
• Techniques for solving equations include: Adding or subtracting the same value from both sides of the equation. Multiplying or dividing both sides of the equation by the same non-zero value. Combining like terms (terms with the same variables and exponents). Isolating the variable on one side of the equation.

Linear Equations

• A linear equation is an equation that can be written in the form ax + b = 0, where 'a' and 'b' are constants, and 'x' is the variable.
• These equations represent a straight line on a graph.
• The slope-intercept form of a linear equation is y = mx + b, where 'm' is the slope and 'b' is the y-intercept.

Polynomials

• A polynomial is an expression consisting of variables and coefficients, involving only the operations of addition, subtraction, and multiplication.
• Examples include: 2x² + 3x - 1, 6y, 5x⁴ + 10x³ - 2x² + 4.
• Polynomials can be classified by the number of terms (monomial, binomial, trinomial, etc.) and by the degree (highest power in the expression).

Factoring

• Factoring is the process of expressing a polynomial as a product of simpler polynomials.
• Factoring can be used to simplify expressions, solve equations, and perform algebraic operations.
• Common factoring techniques include greatest common factor (GCF), difference of squares, and grouping.

Inequalities

• An inequality compares two expressions using symbols like > (greater than), < (less than), ≥ (greater than or equal to), ≤ (less than or equal to), ≠ (not equal to).
• Solving inequalities is similar to solving equations but with a few crucial differences: Multiplying or dividing by a negative number reverses the inequality sign. Graphing solutions on a number line is different from graphing solutions to equations.

Exponents and Radicals

• Exponents represent repeated multiplication. For instance, x² = x * x.
• Radicals represent roots. For instance, √x represents the square root of x.
• Rules for working with these are crucial to simplifying and manipulating expressions.

Quadratic Equations

• A quadratic equation is an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are constants and 'x' is the variable.
• Quadratic equations can be solved using: Factoring, Quadratic formula (−b ± √(b²−4ac))/2a, Completing the square.

Systems of Equations

• Systems of equations involve more than one equation.
• Solving systems usually involves finding a solution (set of values for the unknown variables) that satisfies all of the equations simultaneously.
• Common methods for solving include substitution and elimination.