Solve Quadratic Equations by Completing the Square

Algebra-2

1. Fundamental Concepts

Definition: A quadratic equation is an equation of the form , where .
Completing the Square: A method to solve quadratic equations by transforming them into a perfect square trinomial.
Perfect Square Trinomial: A trinomial that can be factored into the square of a binomial, such as .

2. Key Concepts

General Form:

Completing the Square Steps:

Divide all terms by (the coefficient of ).
Move the constant term to the right side of the equation.
Add to both sides of the equation.
Factor the left side into a perfect square trinomial.
Solve for using the square root property.

Application: Used in various fields such as physics, engineering, and economics to model and solve real-world problems.

3. Examples

Example 1 (Basic)

Problem: Solve by completing the square.

Step-by-Step Solution:

Move the constant term to the right side:
Add to both sides:
Factor the left side:
Take the square root of both sides:
Solve for :
Final solutions: or

Validation: Substitute and into the original equation:
✓

Example 2 (Intermediate)

Problem: Solve by completing the square.

Step-by-Step Solution:

Divide all terms by 2:
Move the constant term to the right side:
Add to both sides:
Factor the left side:
Take the square root of both sides:
Solve for :
Final solutions: or

Validation: Substitute and into the original equation:

✓

4. Problem-Solving Techniques

Visual Strategy: Use a step-by-step checklist to ensure each step is completed correctly.
Error-Proofing: Double-check the addition and subtraction of the constant term to both sides of the equation.
Concept Reinforcement: Practice with different types of quadratic equations to reinforce the steps and improve fluency.