Lesson 36: Quadratic Formula and the Discriminant
The quadratic formula provides a method to solve any quadratic equation of the form ax² + bx + c = 0. The discriminant helps determine the nature of the solutions.
Quadratic Formula
The quadratic formula is:
x = [−b ± √(b² − 4ac)] / (2a)
Steps:
- Identify coefficients a, b, and c in ax² + bx + c = 0
- Compute the discriminant: Δ = b² − 4ac
- Plug values into the formula
- Simplify to find the solutions
The Discriminant
The discriminant, Δ = b² − 4ac, indicates the type of solutions:
- Δ > 0 → Two real and distinct solutions
- Δ = 0 → One real solution (repeated root)
- Δ < 0 → Two complex solutions
Examples
Example 1: Solve x² − 4x − 5 = 0
- a = 1, b = −4, c = −5 → Δ = (−4)² − 4(1)(−5) = 16 + 20 = 36
- x = [4 ± √36]/2 = [4 ± 6]/2
- Solutions: x = 5 or x = −1
Example 2: Solve 2x² + 4x + 2 = 0
- a = 2, b = 4, c = 2 → Δ = 4² − 4(2)(2) = 16 − 16 = 0
- x = [−4 ± √0]/(2*2) = −4/4 = −1
- Solution: x = −1 (repeated root)
Example 3: Solve x² + 2x + 5 = 0
- a = 1, b = 2, c = 5 → Δ = 2² − 4(1)(5) = 4 − 20 = −16
- x = [−2 ± √(−16)]/2 = [−2 ± 4i]/2
- Solutions: x = −1 ± 2i (complex roots)
Practice Problems
- Solve x² + 3x − 10 = 0
- Solve 2x² − 8x + 6 = 0
- Solve x² + 4x + 8 = 0
- Solve 3x² + 5x − 2 = 0
- Solve x² − 6x + 9 = 0