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Questions and Answers
What is the definition of a quadratic equation?
What is the definition of a quadratic equation?
What are the possible values of the discriminant in a quadratic equation?
What are the possible values of the discriminant in a quadratic equation?
What is the purpose of the quadratic formula?
What is the purpose of the quadratic formula?
What can be said about the graph of a quadratic equation?
What can be said about the graph of a quadratic equation?
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What does a negative discriminant indicate about the solutions of a quadratic equation?
What does a negative discriminant indicate about the solutions of a quadratic equation?
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What can be factored about a quadratic equation?
What can be factored about a quadratic equation?
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Study Notes
Quadratic Equations
Definition
- A quadratic equation is a polynomial equation of degree two, which means 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.
Solutions
- A quadratic equation has two solutions, also called roots.
- These solutions can be real or complex numbers.
Methods for Solving Quadratic Equations
1. Factoring
- If the equation can be written in the form: (x - r)(x - s) = 0, then the solutions are x = r and x = s.
- Not all quadratic equations can be factored.
2. Quadratic Formula
- The quadratic formula is: x = (-b ± √(b^2 - 4ac)) / 2a.
- This formula will always give the two solutions to the equation.
- The expression inside the square root (b^2 - 4ac) is called the discriminant.
Discriminant
- If the discriminant is:
- Positive, the equation has two distinct real solutions.
- Zero, the equation has one repeated real solution.
- Negative, the equation has two complex solutions.
Graphing Quadratic Equations
- The graph of a quadratic equation is a parabola that opens upward or downward.
- The x-intercepts of the graph are the solutions to the equation.
- The vertex of the parabola is the minimum or maximum point of the graph.
Quadratic Equations
Definition
- Quadratic equation: a polynomial equation with the highest power of the variable (usually x) being two.
- General form: ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.
Solutions
- A quadratic equation has two solutions, also called roots.
- These solutions can be real or complex numbers.
Methods for Solving Quadratic Equations
Factoring
- If an equation can be written in the form (x - r)(x - s) = 0, then the solutions are x = r and x = s.
- Not all quadratic equations can be factored.
Quadratic Formula
- The quadratic formula is x = (-b ± √(b^2 - 4ac)) / 2a.
- This formula always gives the two solutions to the equation.
- The expression inside the square root (b^2 - 4ac) is called the discriminant.
Discriminant
- If the discriminant is:
- Positive, the equation has two distinct real solutions.
- Zero, the equation has one repeated real solution.
- Negative, the equation has two complex solutions.
Graphing Quadratic Equations
- The graph of a quadratic equation is a parabola that opens upward or downward.
- The x-intercepts of the graph are the solutions to the equation.
- The vertex of the parabola is the minimum or maximum point of the graph.
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Description
Learn about quadratic equations, their definition, and methods for solving them, including factoring and more.
