Solve Quadratic Equations Using Square Roots - Algebra-2

1. Fundamental Concepts

Definition: A quadratic equation is an equation of the form , where .
Square Root Method: This method is used to solve quadratic equations of the form , where is a constant.
Principal Square Root: The principal square root of a number is the non-negative value that, when squared, gives .

2. Key Concepts

Square Root Property: If , then

Isolation of : To use the square root method, isolate on one side of the equation.

Application: This method is particularly useful for solving equations where the variable is squared and there are no other terms involving the variable.

3. Examples

Example 1 (Basic)

Problem: Solve

Step-by-Step Solution:

Apply the square root property:
Simplify:

Validation: Substitute and into the original equation: and ✓

Example 2 (Intermediate)

Problem: Solve

Step-by-Step Solution:

Isolate :
Divide both sides by 2:
Apply the square root property:
Simplify:

Validation: Substitute and into the original equation: and ✓

4. Problem-Solving Techniques

Isolation Technique: Always isolate before applying the square root property.
Check for Extraneous Solutions: After solving, substitute the solutions back into the original equation to ensure they are valid.
Sign Consideration: Remember to include both the positive and negative roots when using the square root property.