Overview
This lecture explains the discriminant in quadratic equations and how it determines the type and nature of solutions.
The Discriminant in Quadratics
- The discriminant helps determine the type of solutions to a quadratic equation.
- Quadratic equations can have 0, 1, or 2 x-intercepts (real solutions).
- The quadratic formula is: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ).
- The discriminant (D) is the part under the square root: ( b^2 - 4ac ).
Analyzing the Discriminant
- If ( D > 0 ) and is a perfect square, there are two real rational roots.
- If ( D > 0 ) but not a perfect square, there are two real irrational roots.
- If ( D = 0 ), there is exactly one real rational root.
- If ( D < 0 ), there are two complex (non-real) roots and no x-intercepts.
Examples and Application
- ( D = 25 ): two real rational roots (e.g., x = 5 and x = -5).
- ( D = 8 ): two real irrational roots (e.g., x = ( \sqrt{8} ) and x = ( -\sqrt{8} )).
- ( D = 0 ): one real rational root (the parabola just touches the x-axis).
- ( D < 0 ): no real roots, only complex solutions.
Key Terms & Definitions
- Discriminant (D) — The expression ( b^2 - 4ac ) in a quadratic equation, used to identify the solution type.
- Rational Number — A number that can be expressed as a fraction of two integers.
- Irrational Number — A number that cannot be written as a simple fraction; its decimal goes on forever without repeating.
- Complex Roots — Solutions that include imaginary numbers and are not on the real number line.
Action Items / Next Steps
- Practice determining the type of solutions by calculating the discriminant for different quadratic equations.
- Review the quadratic formula and its components for upcoming exercises.