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Questions and Answers
What is the highest power of the variable in a quadratic equation?
What is the highest power of the variable in a quadratic equation?
What is the name of the formula used to find the roots of a quadratic equation?
What is the name of the formula used to find the roots of a quadratic equation?
What is the name of the graph of a quadratic equation?
What is the name of the graph of a quadratic equation?
What is the value of the discriminant if a quadratic equation has one repeated real root?
What is the value of the discriminant if a quadratic equation has one repeated real root?
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What is the name of the vertical line passing through the vertex of a parabola?
What is the name of the vertical line passing through the vertex of a parabola?
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What is the term for the minimum or maximum point of a parabola?
What is the term for the minimum or maximum point of a parabola?
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Study Notes
Definition and Forms
- A quadratic equation is a polynomial equation of degree two, meaning the highest power of the variable (usually x) is two.
- The general form of a quadratic equation is: ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
Types of Quadratic Equations
- Monic quadratic equation: a = 1, e.g., x^2 + bx + c = 0
- Non-monic quadratic equation: a ≠ 1, e.g., 2x^2 + bx + c = 0
Solutions of Quadratic Equations
- Quadratic Formula: x = (-b ± √(b^2 - 4ac)) / 2a, used to find the roots of the equation.
- Factoring: if the equation can be written in the form (x - r)(x - s) = 0, then the roots are x = r and x = s.
Graphs of Quadratic Equations
- Parabola: the graph of a quadratic equation, opening upward or downward.
- Vertex: the minimum or maximum point of the parabola.
- Axis of Symmetry: the vertical line passing through the vertex.
Key Features
-
Discriminant: b^2 - 4ac, determines the nature of the roots:
- Positive: two distinct real roots
- Zero: one repeated real root
- Negative: two complex conjugate roots
- x-intercepts: the roots of the equation, where the graph crosses the x-axis.
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Description
Learn the basics of quadratic equations, including definition, types, solutions, graphs, and key features. Understand the quadratic formula, factoring, and how to analyze the graphs of quadratic equations.
