Overview
This course covers key Algebra 1 foundations: variables and expressions, equations and their properties, linear equations, inequalities, systems, polynomials, exponents, factoring, rational expressions, roots and radicals, quadratic equations, and applications like word problems and graphing.
Variables & Expressions
- A variable is a symbol, usually a lowercase letter, to represent an unknown value.
- Expressions are combinations of numbers, variables, and operations, e.g., 2x + 3.
- A term is a single number, a variable, or a product of numbers and variables.
- A coefficient is the number multiplying a variable in a term; a constant stands alone.
- Algebraic expressions can be simplified or evaluated by substituting values for variables.
Equations & Properties
- An equation states that two expressions are equal, using the = sign.
- To solve equations, use properties: addition/subtraction (add/subtract same value both sides), multiplication/division (multiply/divide both sides by same non-zero number).
- The goal is to isolate the variable on one side (e.g., x = ...).
- Use the distributive property (a(b + c) = ab + ac) to expand or simplify.
Linear Equations & Inequalities
- Linear equations in one variable look like ax + b = c, a ≠ 0.
- Solving steps: simplify both sides, isolate variable term, then isolate variable.
- Combine like terms (same variable(s) to the same power).
- Inequalities use <, >, ≤, or ≥; solve similarly to equations, but reverse inequality when multiplying/dividing by a negative.
Systems of Equations
- A system contains two or more equations with the same variables.
- Solution methods: graphing (intersection point), substitution, or elimination.
- Solutions can be one point, none (parallel lines), or infinitely many (same line).
Polynomials, Exponents & Factoring
- A polynomial is a sum of terms with non-negative integer exponents.
- Add/subtract polynomials by combining like terms; multiply using distributive property, FOIL, or special products.
- Exponent rules: xᵃ × xᵇ = xᵃ⁺ᵇ, (xᵃ)ᵇ = xᵃᵇ, x⁰ = 1, x⁻ⁿ = 1/xⁿ.
- Factorization rewrites a polynomial as a product of simpler polynomials using GCF, grouping, or special patterns.
Rational Expressions
- Rational expressions are ratios of polynomials; undefined where denominator = 0.
- Simplify by factoring and canceling common factors.
- To add/subtract, use a common denominator (preferably the LCD).
- Rationalize denominators by multiplying by a form of 1 to remove radicals.
Roots, Radicals, & Rational Exponents
- The square root of x is a number y with y² = x; principal root is positive; negative roots by ±.
- To simplify roots, factor the radicand for perfect squares/cubes, etc.
- Fractional exponents: xᵐ⁄ⁿ = n-th root of (x^m), or (n-th root of x) to the m.
- Rationalize and combine radicals as with like terms.
Quadratic Equations
- Standard form: ax² + bx + c = 0; a ≠ 0.
- Solving methods: factoring, square root property, completing the square, or quadratic formula.
- Quadratic formula: x = [-b ± √(b²-4ac)]/(2a).
- The discriminant (b²–4ac) indicates solution type: >0 (2 real), =0 (1 real), <0 (no real).
Graphing & Coordinate Geometry
- Ordered pairs (x, y) locate points; the coordinate plane has 4 quadrants.
- The graph of a linear equation is a straight line; the slope (m) = rise/run.
- Equation forms: y = mx + b (slope-intercept), Ax + By = C (standard).
- Parallel lines: equal slopes; perpendicular: slopes are negative reciprocals.
- Distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²].
Applications & Word Problems
- Read carefully—assign variables to unknowns; write equations based on relationships.
- For mixture, motion, and work problems, use the right formulas (e.g., d = rt, work = rate × time).
- Set up equations and solve; check if answers make sense for the context.
Key Terms & Definitions
- Variable — symbol representing an unknown value (e.g., x)
- Coefficient — number multiplying a variable in a term
- Constant — number alone in an expression/equation
- Term — a number, variable, or their product
- Expression — combination of terms using operations
- Equation — statement that two expressions are equal
- Inequality — compares expressions using <, >, ≤, ≥
- Polynomial — sum of terms with non-negative integer exponents
- Quadratic Equation — equation of the form ax² + bx + c = 0
- Rational Expression — ratio of two polynomials
- Radical — square root, cube root, etc., symbol
- Discriminant — b²–4ac in the quadratic formula; tells solution type
- Slope — change in y over change in x for a line
Action Items / Next Steps
- Review your textbook's practice problems for each topic.
- Memorize key formulas (e.g., quadratic formula, exponent rules).
- Practice factoring, solving equations, and graphing.
- Work on word problems to apply concepts.
- Bring any confusing homework or test prep questions to your teacher or tutor for clarification.
Tip: For exams, always check if your answers make sense and are properly simplified or rationalized as required.