Algebra Fundamentals

Fundamental Concepts

• Algebra is a branch of mathematics that uses symbols to represent numbers and quantities, and the relationships among them.
• It uses letters and symbols to create equations and formulas, which allow for generalizations about mathematical relationships and solutions to a wide variety of problems.
• It's a powerful tool for modeling real-world situations.
• Variables represent unknown values.
• Constants represent fixed values.
• Expressions combine variables, constants and operations (addition, subtraction, multiplication, division, etc.).

Basic Operations

• Addition: Combining quantities. Commutative (order doesn't matter) and associative (grouping doesn't matter).
• Subtraction: Finding the difference between quantities. Not commutative.
• Multiplication: Repeated addition. Commutative and associative.
• Division: Finding how many times one quantity is contained within another. Not commutative.
• Order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (left to right), Addition and Subtraction (left to right). This ensures a consistent solution method.

Equations

• An equation is a statement that two expressions are equal.
• Solving equations involves manipulating both sides of the equation to isolate the unknown variable.
• Key properties of equality: If a=b, then a+c=b+c, a-c=b-c, ac=bc, a/c=b/c (provided c ≠0).
• Techniques for solving equations include adding, subtracting, multiplying or dividing both sides of the equation by the same value.
• The solution to an equation is the value (or values) that make the equation true.

Inequalities

• An inequality is a mathematical statement that compares two expressions using symbols like < (less than), > (greater than), ≤ (less than or equal to), ≥ (greater than or equal to), or ≠ (not equal to).
• Solving inequalities is similar to solving equations, but the direction of the inequality sign changes when multiplying or dividing both sides by a negative number.

Linear Equations

• A linear equation has the general form ax + b = c
• 'x' is the variable
• 'a' and 'b' and 'c' are constants.
• The graph of a linear equation is a straight line.
• Solutions to linear equations often involve isolating the variable on one side of the equation.
• Finding the slope of a linear equation
• Finding the y-intercept of a linear equation
• Forms of linear equations, including slope-intercept form (y = mx + b) and point-slope form.

Polynomials

• Polynomials are algebraic expressions consisting of variables and coefficients, combined using only addition, subtraction, and multiplication.
• Different degrees of polynomials – linear (degree 1), quadratic (degree 2), cubic (degree 3), etc. – dictate different levels of complexity and solution methods.
• Operations on polynomials (addition, subtraction, multiplication).
• Factoring polynomials for simplifying solutions.

Factoring

• Factoring is the process of finding the factors of a mathematical expression.
• Factoring can simplify expressions for easier operations or solutions.

Systems of Equations

• Systems of equations consist of two or more equations that have the same variables.
• Solving methods:
  • substitution
  • elimination.
• Finding the point of intersection of graphed equations.

Exponents and Radicals

• Exponents represent repeated multiplication.
• Rules of exponents, including negative exponents, fractional exponents and simplifying powers.
• Radicals represent roots (square roots, cube roots, etc.)

Quadratic Equations

• Quadratic equations have the general form ax² + bx + c = 0
• Solving quadratic equations using factoring, completing the square, or the quadratic formula.
• Finding solutions called roots, or x-intercepts of the function represented by the equation.

This quiz covers the fundamental concepts of algebra, including the use of symbols, variables, and constants. Students will explore basic operations such as addition, subtraction, multiplication, and division, along with the importance of order of operations. Ideal for those looking to enhance their understanding of algebraic principles.