How to Determine Solutions of Quadratic Equations: A Step-by-Step Guide

Understanding how to determine if a value is a solution to a quadratic equation is an essential skill in algebra. A quadratic equation typically appears in the form of ax² + bx + c = 0, where a, b, and c are constants. For our example, the equation is x² + 3x + 2 = 0.

To check if a specific value of x is a solution, follow these steps:

1. **Substitution**: Replace x in the equation with the value you want to test. For instance, let’s check two values: -2 and 5.

2. **Simplification**: After substituting, simplify the equation. This involves performing the operations in the equation to see if the left-hand side equals zero.

3. **Conclusion**: If the result of the simplification is zero, then the value is a solution to the equation. If it is not zero, then the value is not a solution.

Let’s go through both values step-by-step:

- **Testing x = -2**:
- Substitute -2 into the equation: (-2)² + 3(-2) + 2
- Simplify: 4 - 6 + 2 = 0
- Since the result is zero, -2 is indeed a solution to the equation.

- **Testing x = 5**:
- Substitute 5 into the equation: (5)² + 3(5) + 2
- Simplify: 25 + 15 + 2 = 42
- Since the result is not zero, 5 is not a solution.

This method can be applied to any quadratic equation. It's important to practice with different equations and values to become proficient. Quadratic equations often arise in various real-world applications, such as physics when calculating projectile motion, optimizing areas, or in finance when determining profit maximization.

By mastering substitution and simplification, you will enhance your problem-solving skills in algebra and prepare yourself for more advanced mathematical concepts. Don't hesitate to explore more problems and practice regularly to build your confidence. Check the equations section for additional formulas and examples to further enhance your understanding.