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What does algebra deal with?
What is a variable in algebra?
What is an algebraic expression?
What is an equation in algebra?
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What is a quadratic equation?
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What is the purpose of adding or subtracting the same value to both sides of an equation?
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What is the name of the formula used to solve quadratic equations?
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What is the coordinate plane used for in algebra?
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Study Notes
Algebra
Definition
- Algebra is a branch of mathematics that deals with the study of variables and their relationships, often expressed through the use of symbols, equations, and functions.
Key Concepts
- Variables: Letters or symbols that represent unknown values or quantities.
- Constants: Numbers that do not change value.
- Algebraic expressions: Combinations of variables, constants, and mathematical operations.
- Equations: Statements that two algebraic expressions are equal.
- Functions: Relations between variables, often expressed as f(x) = ...
Types of Equations
-
Linear equations: Equations in which the highest power of the variable is 1.
- Example: 2x + 3 = 5
-
Quadratic equations: Equations in which the highest power of the variable is 2.
- Example: x^2 + 4x + 4 = 0
-
Polynomial equations: Equations involving variables and coefficients, with only positive integer exponents.
- Example: x^3 + 2x^2 - 7x - 12 = 0
Solving Equations
- Addition and subtraction: Isolate the variable by adding or subtracting the same value to both sides of the equation.
- Multiplication and division: Isolate the variable by multiplying or dividing both sides of the equation by the same value.
- Factoring: Expressing an equation as a product of simpler expressions, often used to solve quadratic equations.
- Quadratic formula: A formula used to solve quadratic equations, given by x = (-b ± √(b^2 - 4ac)) / 2a.
Graphing
- Coordinate plane: A graph with two axes (x and y) used to plot points and graph functions.
- Graphing linear equations: Plotting points on the coordinate plane to visualize the equation.
- Graphing quadratic equations: Plotting points on the coordinate plane to visualize the equation, often resulting in a parabola.
Functions
- Domain: The set of input values for which the function is defined.
- Range: The set of output values of the function.
- Composition of functions: Combining two or more functions to create a new function.
- Inverse functions: Functions that "undo" the effect of another function.
Algebra
Definition
- A branch of mathematics dealing with variables and their relationships, expressed through symbols, equations, and functions.
Key Concepts
- Variables: Represent unknown values or quantities.
- Constants: Numbers that do not change value.
- Algebraic expressions: Combinations of variables, constants, and mathematical operations.
- Equations: Statements that two algebraic expressions are equal.
- Functions: Relations between variables, often expressed as f(x) =...
Types of Equations
Linear Equations
- Highest power of the variable is 1.
- Example: 2x + 3 = 5
Quadratic Equations
- Highest power of the variable is 2.
- Example: x^2 + 4x + 4 = 0
Polynomial Equations
- Equations involving variables and coefficients, with only positive integer exponents.
- Example: x^3 + 2x^2 - 7x - 12 = 0
Solving Equations
- Addition and subtraction: Isolate the variable by adding or subtracting the same value to both sides of the equation.
- Multiplication and division: Isolate the variable by multiplying or dividing both sides of the equation by the same value.
- Factoring: Expressing an equation as a product of simpler expressions, often used to solve quadratic equations.
- Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a, used to solve quadratic equations.
Graphing
- Coordinate plane: A graph with two axes (x and y) used to plot points and graph functions.
- Graphing linear equations: Plotting points on the coordinate plane to visualize the equation.
- Graphing quadratic equations: Plotting points on the coordinate plane to visualize the equation, often resulting in a parabola.
Functions
- Domain: The set of input values for which the function is defined.
- Range: The set of output values of the function.
- Composition of functions: Combining two or more functions to create a new function.
- Inverse functions: Functions that "undo" the effect of another function.
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Description
Learn the fundamentals of algebra, including variables, constants, algebraic expressions, and equations. Understand the core concepts of this branch of mathematics.
