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Questions and Answers
What is an algebraic expression?
What is an algebraic expression?
What is the term for an algebraic expression with a single term?
What is the term for an algebraic expression with a single term?
How do you combine like terms in algebraic expressions?
How do you combine like terms in algebraic expressions?
What is the distributive property used for in algebraic expressions?
What is the distributive property used for in algebraic expressions?
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What is the correct order of operations when evaluating algebraic expressions?
What is the correct order of operations when evaluating algebraic expressions?
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What is the first step in simplifying an algebraic expression?
What is the first step in simplifying an algebraic expression?
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What is the primary condition for an algebraic expression to be a quadratic equation?
What is the primary condition for an algebraic expression to be a quadratic equation?
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What is the name of the method that expresses the quadratic equation as a product of two binomials and solves for x?
What is the name of the method that expresses the quadratic equation as a product of two binomials and solves for x?
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What is the term for the value of b^2 - 4ac in the Quadratic Formula?
What is the term for the value of b^2 - 4ac in the Quadratic Formula?
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What is the interpretation of a negative value of the discriminant in the Quadratic Formula?
What is the interpretation of a negative value of the discriminant in the Quadratic Formula?
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Which type of quadratic equation has a coefficient of a equal to 1?
Which type of quadratic equation has a coefficient of a equal to 1?
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What is a real-world application of quadratic equations?
What is a real-world application of quadratic equations?
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Study Notes
Algebraic Expressions
Definition
- An algebraic expression is a mathematical phrase that combines numbers, variables, and mathematical operations.
- It can be a single term or a combination of terms separated by operators (+, -, x, /).
Types of Algebraic Expressions
- Monomials: Expressions with a single term, e.g., 2x, 5y, 3z
- Binomials: Expressions with two terms, e.g., 2x + 3, x^2 - 4
- Polynomials: Expressions with three or more terms, e.g., x^2 + 3x - 2, 2x^3 - 5x^2 + x - 1
Operations with Algebraic Expressions
- Addition and Subtraction: Combine like terms by adding or subtracting their coefficients.
- Multiplication: Multiply each term in one expression by each term in the other expression.
- Distributive Property: Expand products by multiplying each term in the parentheses by the factor outside, e.g., 2(x + 3) = 2x + 6
Simplifying Algebraic Expressions
- Combine like terms: Combine terms with the same variable(s) and coefficient(s).
- Remove parentheses: Use the distributive property to expand products.
- Combine constants: Combine numeric terms.
Evaluating Algebraic Expressions
- Replace variables with given values to evaluate the expression.
- Follow the order of operations (PEMDAS): parentheses, exponents, multiplication and division, addition and subtraction.
Algebraic Expressions
- An algebraic expression is a mathematical phrase combining numbers, variables, and mathematical operations.
- It can be a single term or a combination of terms separated by operators (+, -, x, /).
Types of Algebraic Expressions
- Monomials are expressions with a single term, such as 2x, 5y, or 3z.
- Binomials are expressions with two terms, such as 2x + 3 or x^2 - 4.
- Polynomials are expressions with three or more terms, such as x^2 + 3x - 2 or 2x^3 - 5x^2 + x - 1.
Operations with Algebraic Expressions
- To add or subtract algebraic expressions, combine like terms by adding or subtracting their coefficients.
- To multiply algebraic expressions, multiply each term in one expression by each term in the other expression.
- The distributive property allows expanding products by multiplying each term in the parentheses by the factor outside, e.g., 2(x + 3) = 2x + 6.
Simplifying Algebraic Expressions
- Combine like terms to simplify algebraic expressions.
- Remove parentheses by using the distributive property to expand products.
- Combine constants by adding or subtracting numeric terms.
Evaluating Algebraic Expressions
- Replace variables with given values to evaluate the expression.
- Follow the order of operations (PEMDAS): parentheses, exponents, multiplication and division, addition and subtraction.
Quadratic Equations
- A quadratic equation is a type of algebraic expression that can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
- The highest power of the variable (x) in a quadratic equation is 2.
Key Characteristics
- The graph of a quadratic equation is a parabola that opens upwards or downwards.
- Quadratic equations can be factored into the product of two binomials.
Types of Quadratic Equations
- Monic quadratic equations have a = 1, such as x^2 + 5x + 6 = 0.
- Non-monic quadratic equations have a ≠ 1, such as 2x^2 + 3x + 1 = 0.
Methods for Solving Quadratic Equations
- Factoring involves expressing the equation as a product of two binomials and solving for x.
- The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
- Graphing involves finding the x-intercepts of the parabola by graphing the equation.
Quadratic Formula
- The discriminant, b^2 - 4ac, determines the number of solutions.
- If b^2 - 4ac > 0, there are two distinct real solutions.
- If b^2 - 4ac = 0, there is one real solution (repeated root).
- If b^2 - 4ac < 0, there are no real solutions (complex solutions).
Applications of Quadratic Equations
- Quadratic equations can be used to model projectile motion.
- They can be used to solve optimization problems, finding the maximum or minimum value of a quadratic function.
- Quadratic equations are applied in physics and engineering to describe the motion of objects and systems.
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Description
Learn about algebraic expressions, including monomials, binomials, and polynomials, and how they combine numbers, variables, and mathematical operations.
