Algebra Fundamentals

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols to solve equations.

Variables: Symbols (often letters) used to represent unknown values.
Constants: Fixed values that do not change.
Coefficients: Numerical factors multiplying variables (e.g., in 3x, 3 is the coefficient).
Expressions: Combinations of variables, constants, and operators (e.g., 2x + 3).
Equations: Mathematical statements asserting equality between expressions (e.g., 2x + 3 = 7).

Linear Equations: Equations of the form ax + b = c. Solve for x.
- Techniques: Isolation of variable, using inverse operations.
Quadratic Equations: Equations of the form ax² + bx + c = 0.
- Methods: Factoring, completing the square, quadratic formula (x = [-b ± √(b²-4ac)]/(2a)).
Inequalities: Expressions showing the relationship between quantities that are not necessarily equal.
- Solving involves similar techniques, but requires flipping the inequality sign when multiplying/dividing by a negative.

Definition: A relation where each input has a unique output.
Notation: f(x) denotes a function of x.
Types:
- Linear Function: Graphs as straight lines (y = mx + b).
- Quadratic Function: Parabolas (y = ax² + bx + c).

Definition: An algebraic expression with multiple terms (e.g., 4x³ + 3x² - x + 2).
Degree: Highest power of the variable; determines the shape of the graph.
Operations: Addition, subtraction, multiplication, and division (synthetic division).

Purpose: Breaking down expressions into products of simpler factors.
Techniques:
- Common factor extraction.
- Grouping.
- Special products (e.g., difference of squares, perfect square trinomials).

Check Solutions: Always substitute back into original equations.
Use Graphing: Visual representation helps in understanding relationships and solutions.
Practice: Regular solving of problems enhances skills and familiarity with concepts.

Definition: Algebra is a branch of mathematics dealing with symbols and their manipulation to solve equations.
Variables: Letters used to represent unknown values, they are flexible and can change.
Constants: Fixed values that remain unchanged throughout a problem.
Coefficients: Numbers that multiply variables in expressions.
Expressions: Combinations of variables, constants, and operations like addition, subtraction, multiplication, and division.
Equations: Statements that show equality between two or more expressions.

Linear Equations: Equations like ax + b = c, where x is the unknown variable and a, b, and c are constants.
Solving Linear Equations:
- Isolate the variable (x) by performing operations on both sides of the equation until only x remains.
- Use inverse operations to counteract the operations applied to x.
Quadratic Equations: Equations of the form ax² + bx + c = 0, where x is the unknown variable, and a, b, and c are constants.
Solving Quadratic Equations:
- Factoring: Break the equation into simpler expressions that multiply to the original equation.
- Completing the Square: Manipulating the equation to create a perfect square trinomial, then using the square root property.
- Quadratic Formula: Applying the formula x = [-b ± √(b²-4ac)]/(2a).
Inequalities: Mathematical statements where two expressions are not necessarily equal, instead, they express relationships like "greater than," "less than," or "greater than or equal to."
Solving Inequalities:
- Utilize similar techniques to solving equations.
- Remember to flip the inequality sign when you multiply or divide both sides by a negative number.

Definition: A relation where each input has a unique output.
Notation: Typically represented by f(x), where x is the input.
Types:
- Linear Functions: Represented by straight lines on a graph (y = mx + b).
- Quadratic Functions: Represented by parabolas on a graph (y = ax² + bx + c).

Definition: An algebraic expression with multiple terms.
Degree: The highest power of the variable, it determines the shape of the graph.
Operations:
- Addition: Combine the terms with the same variable and power.
- Subtraction: Combine terms with the same variable and power, subtracting coefficients.
- Multiplication: Multiply each term in one polynomial by each term in the other.
- Division: Divide two polynomials using methods like synthetic division.

Purpose: To break down expressions into a product of simpler factors.
Techniques:
- Common Factor Extraction: Find the greatest common factor among all terms and take it out of the expression.
- Grouping: Group terms to find a common factor within the groups.
- Special Products: Use identities for difference of squares, perfect square trinomials, etc.

Distance Formula: Find the distance between two points on a coordinate plane.
- d = √((x2 - x1)² + (y2 - y1)²).
Slope Formula: Determine the slope of a line (inclination).
- m = (y2 - y1) / (x2 - x1).
Point-Slope Form: Find the equation of a line using a point and its slope.
- y - y1 = m(x - x1).

Problem-solving in various Fields: Finance, engineering, science, and more.
Modeling Real-World Situations: Use equations and functions to describe real-world scenarios.

Check Solutions: Substitute your solution back into the original equation to verify its correctness.
Use Graphing: Visual representation can help understand relationships and solutions.
Practice: Regularly solve problems to improve skills and familiarity with concepts.