Key Concepts:
- Quadratic Formula: (x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}), used when factoring is difficult.
- Discriminant \Delta Analysis: Determines the number of real solutions based on (b^2 - 4ac):
- (\Delta > 0): Two real solutions
- (\Delta = 0): One real solution
- (\Delta < 0): No real solutions
- Vieta's Formulas: The sum of the solutions to a quadratic equation can be expressed as (-\frac{b}{a}), the product of the solutions can be expressed as (\frac{c}{a}).
Skills Covered:
- Using the quadratic formula
- Applying the discriminant to determine the number of solutions
- Applying Vieta's formulas to determine the sum or product of solutions
Example Problems:
- Solve (2x^2 - 3x - 5 = 0) using the quadratic formula.
- Determine how many real solutions exist for (3x^2 + 4x + 11 = 0).
- Find the sum of solutions to the equation (2x^2 + 5x + 1 = 0).