Solve Quadratic Equations by Using the Quadratic Formula - Algebra-2

1. Fundamental Concepts

Definition: The quadratic formula is used to solve any quadratic equation of the form where .
Formula: The solutions to the quadratic equation are given by
Discriminant: The discriminant, , determines the nature of the roots:
- If , there are two distinct real roots.
- If , there is one real root (a repeated root).
- If , there are two complex conjugate roots.

Standard Form:

Quadratic Formula:

Application: Used in physics, engineering, and economics to model parabolic motion and other phenomena.

Problem: Solve the quadratic equation

Validation: Substitute and into the original equation:

- For :

Problem: Solve the quadratic equation

Validation: Substitute and into the original equation:

- For :

Identify Coefficients: Clearly identify , , and from the standard form of the quadratic equation.
Calculate the Discriminant: Determine the nature of the roots by calculating .
Apply the Quadratic Formula: Use the quadratic formula to find the roots, ensuring to handle the sign correctly.
Check Solutions: Always substitute the solutions back into the original equation to verify their correctness.