Algebra Basics Quiz

Algebra

Basics of Algebra

• Variables: Symbols (e.g., x, y) used to represent unknown values.
• Constants: Fixed values (e.g., numbers like 2, -5).
• Expressions: Combinations of variables and constants (e.g., 3x + 2).
• Equations: Statements that two expressions are equal (e.g., 2x + 3 = 7).

Operations

• Addition/Subtraction: Combining like terms (e.g., 2x + 3x = 5x).
• Multiplication: Distributing (e.g., a(b + c) = ab + ac).
• Division: Simplifying fractions (e.g., x²/x = x).

Solving Equations

• One-step Equations: Isolate the variable (e.g., x + 3 = 5 → x = 2).
• Two-step Equations: Perform two operations (e.g., 2x + 3 = 7 → 2x = 4 → x = 2).
• Multi-step Equations: Combine steps as needed to isolate the variable.

Types of Equations

• Linear Equations: Form y = mx + b; graph as a straight line.
• Quadratic Equations: Form ax² + bx + c = 0; solutions via factoring, completing the square, or the quadratic formula.
• Polynomial Equations: Involve variables raised to powers (e.g., x^3 + 4x^2 - x + 6 = 0).

Functions

• Definition: A relation that assigns exactly one output for each input.
• Notation: f(x) denotes a function of x.
• Types: Linear, quadratic, exponential, etc.

Factoring

• Common Methods:
  • GCF (Greatest Common Factor): Factor out the largest common factor.
  • Difference of Squares: a² - b² = (a - b)(a + b).
  • Trinomials: ax² + bx + c factored into (px + q)(rx + s).

Inequalities

• Symbols: < (less than), > (greater than), ≤ (less than or equal), ≥ (greater than or equal).
• Solving: Similar to equations; maintain inequality direction when multiplying/dividing by negative numbers.

Graphing

• Coordinate Plane: Consists of x (horizontal) and y (vertical) axes.
• Plotting Points: (x, y) represents a point on the graph.
• Slope: m = (y2 - y1) / (x2 - x1); indicates steepness of a line.

Systems of Equations

• Definition: Two or more equations with the same set of variables.
• Methods of Solving:
  • Graphing: Finding intersection points.
  • Substitution: Solve one equation for a variable and substitute.
  • Elimination: Add or subtract equations to eliminate a variable.

Key Concepts

• Exponents: ( x^n ) denotes x multiplied by itself n times.
• Radicals: √x denotes the square root of x.
• Absolute Value: |x| represents the distance of x from zero on the number line.

Basics of Algebra

• Variables represent unknowns using symbols such as x and y.
• Constants are fixed numerical values, like 2 or -5.
• Expressions are formed by combining variables and constants, exemplified as 3x + 2.
• Equations establish equality between two expressions, such as in 2x + 3 = 7.

Operations

• Addition and subtraction involve combining like terms, e.g., 2x + 3x results in 5x.
• Multiplication uses distribution to expand expressions, demonstrated by a(b + c) = ab + ac.
• Division allows for simplification of fractions, such as x²/x = x.

Solving Equations

• One-step equations require isolating the variable directly, illustrated by x + 3 = 5 leading to x = 2.
• Two-step equations involve two operations, e.g., 2x + 3 = 7 simplifies to 2x = 4 and then x = 2.
• Multi-step equations combine various operations to isolate the variable as needed.

Types of Equations

• Linear equations are expressed in the form y = mx + b and graph as straight lines.
• Quadratic equations follow the standard ax² + bx + c = 0, solvable via factoring, completing the square, or using the quadratic formula.
• Polynomial equations feature variables raised to powers, such as x^3 + 4x^2 - x + 6 = 0.

Functions

• A function is defined as a relation that assigns exactly one output for each specific input.
• Function notation is represented as f(x), indicating the function's output relative to input x.
• Various types include linear, quadratic, and exponential functions.

Factoring

• The Greatest Common Factor (GCF) involves extracting the largest factor common to terms.
• The difference of squares formula states that a² - b² can be factored into (a - b)(a + b).
• Trinomials can be factored into the product of two binomials, exemplified as ax² + bx + c = (px + q)(rx + s).

Inequalities

• Common inequality symbols include < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to).
• Solving inequalities is similar to solving equations, but it is crucial to maintain the inequality direction when multiplying or dividing by negative numbers.

Graphing

• The coordinate plane consists of two axes: the x-axis (horizontal) and the y-axis (vertical).
• Each point on the graph is represented by coordinates (x, y).
• The slope of a line is calculated using the formula m = (y2 - y1) / (x2 - x1), indicating the line's steepness.

Systems of Equations

• A system of equations consists of two or more equations sharing the same variables.
• Methods to solve systems include:
  • Graphing to find intersection points visually.
  • Substitution, where one equation is solved for a variable and substituted into another.
  • Elimination, which involves adding or subtracting equations to remove a variable.

Key Concepts

• Exponents denote repeated multiplication of a number, represented as ( x^n ).
• Radicals represent the square root of a number, shown as √x.
• Absolute value |x| signifies the distance of x from zero on the number line.