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Questions and Answers
What is the general form of an algebraic equation?
What is the degree of the quadratic equation x^2 + 4x + 4 = 0?
What method can be used to solve the equation x^2 + 5x + 6 = 0?
What property allows us to add or subtract equations as long as they have the same variables and coefficients?
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What is the formula used to solve quadratic equations?
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What type of equation is x^3 - 2x^2 - 5x + 1 = 0?
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Study Notes
Algebraic Equations
Definition
- An algebraic equation is an equation involving variables and constants, and only uses addition, subtraction, multiplication, and division.
- It is written in the form: ax^n + bx^(n-1) + ... + cx + d = 0, where a, b, ..., c, d are constants and x is the variable.
Types of Algebraic Equations
- Linear Equations: Degree of the equation is 1 (e.g. 2x + 3 = 5)
- Quadratic Equations: Degree of the equation is 2 (e.g. x^2 + 4x + 4 = 0)
- Cubic Equations: Degree of the equation is 3 (e.g. x^3 - 2x^2 - 5x + 1 = 0)
- Polynomial Equations: Degree of the equation is greater than 3 (e.g. x^4 - 3x^3 + 2x^2 - x + 1 = 0)
Solving Algebraic Equations
- Factoring: Expressing the equation as a product of simpler expressions (e.g. x^2 + 5x + 6 = (x + 3)(x + 2) = 0)
- Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a, used to solve quadratic equations
- Graphical Method: Solving equations by finding the points of intersection between the graph of the equation and the x-axis
Properties of Algebraic Equations
- Addition and Subtraction: Equations can be added or subtracted, as long as they have the same variables and coefficients.
- Multiplication and Division: Equations can be multiplied or divided, as long as they have the same variables and coefficients.
- Distributive Property: a(b + c) = ab + ac, used to expand equations.
Algebraic Equations
- Algebraic equations involve variables and constants, and only use addition, subtraction, multiplication, and division.
Definition and Form
- The general form of an algebraic equation is ax^n + bx^(n-1) +...+ cx + d = 0.
- In this form, a, b,..., c, d are constants, and x is the variable.
Types of Algebraic Equations
- Linear Equations: Have a degree of 1, such as 2x + 3 = 5.
- Quadratic Equations: Have a degree of 2, such as x^2 + 4x + 4 = 0.
- Cubic Equations: Have a degree of 3, such as x^3 - 2x^2 - 5x + 1 = 0.
- Polynomial Equations: Have a degree greater than 3, such as x^4 - 3x^3 + 2x^2 - x + 1 = 0.
Solving Algebraic Equations
- Factoring: Involves expressing the equation as a product of simpler expressions, such as x^2 + 5x + 6 = (x + 3)(x + 2) = 0.
- Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a, used to solve quadratic equations.
- Graphical Method: Solves equations by finding the points of intersection between the graph of the equation and the x-axis.
Properties of Algebraic Equations
- Addition and Subtraction: Equations can be added or subtracted if they have the same variables and coefficients.
- Multiplication and Division: Equations can be multiplied or divided if they have the same variables and coefficients.
- Distributive Property: a(b + c) = ab + ac, used to expand equations.
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Description
Learn about algebraic equations, including linear and quadratic equations, and their definitions and types. Understand the forms of equations and how to work with them.
