Overview
This lecture explains the discriminant in quadratic equations and how it indicates the type and number of solutions a quadratic equation has.
Discriminant and Types of Solutions
- The discriminant helps determine what kind of solutions a quadratic equation will have.
- A quadratic equation can have one, none, or two x-intercepts (real roots).
- The discriminant is found in the quadratic formula, specifically under the square root.
- The discriminant (D) is calculated by D = b² - 4ac.
- The value of D tells us the nature of the roots without solving the entire equation.
Cases Based on Discriminant Value
- If D is positive and a perfect square (e.g., 1, 4, 9...), there are two real rational roots.
- If D is positive but not a perfect square (e.g., 8, 7, 11...), there are two real irrational roots.
- If D = 0, there is one real rational root (a repeated root).
- If D is negative, there are two complex roots (no real x-intercepts).
Key Terms & Definitions
- Quadratic Equation — An equation in the form ax² + bx + c = 0.
- Discriminant (D) — The expression b² - 4ac used to determine root types.
- Rational Root — A root that can be expressed as a ratio of two integers.
- Irrational Root — A root that cannot be written as a simple fraction; non-repeating and non-terminating decimal.
- Complex Roots — Roots that include an imaginary number; occur when discriminant is negative.
Action Items / Next Steps
- Practice determining the type of solutions for given quadratic equations by calculating the discriminant.