Solve Quadratic Equations by Completing the Square - Algebra-2

1. Fundamental Concepts

Definition: A quadratic equation is an equation of the form $$ax^2 + bx + c = 0\$$ , where $$a \neq 0\$$ .
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 $$x^2 + 6x + 9 = (x + 3)^2\$$ .

2. Key Concepts

General Form: $$ax^2 + bx + c = 0\$$

Completing the Square Steps:

Divide all terms by $$a\$$ (the coefficient of $$x^2\$$ ).
Move the constant term to the right side of the equation.
Add $$\left(\frac{b}{2a}\right)^2\$$ to both sides of the equation.
Factor the left side into a perfect square trinomial.
Solve for $$x\$$ 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 $$x^2 + 6x + 5 = 0\$$ by completing the square.

Step-by-Step Solution:

Move the constant term to the right side: $$x^2 + 6x = -5\$$
Add $$\left(\frac{6}{2}\right)^2 = 9\$$ to both sides: $$x^2 + 6x + 9 = 4\$$
Factor the left side: $$ (x + 3)^2 = 4 \$$
Take the square root of both sides: $$ x + 3 = \pm 2 \$$
Solve for $$x\$$ : $$ x = -3 \pm 2 \$$
Final solutions: $$ x = -1 \$$ or $$ x = -5 \$$

Validation: Substitute $$x = -1\$$ and $$x = -5\$$ into the original equation: $$ (-1)^2 + 6(-1) + 5 = 1 - 6 + 5 = 0 \$$
$$ (-5)^2 + 6(-5) + 5 = 25 - 30 + 5 = 0 \$$ ✓

Example 2 (Intermediate)

Problem: Solve $$2x^2 + 8x - 10 = 0\$$ by completing the square.

Step-by-Step Solution:

Divide all terms by 2: $$x^2 + 4x - 5 = 0\$$
Move the constant term to the right side: $$x^2 + 4x = 5\$$
Add $$\left(\frac{4}{2}\right)^2 = 4\$$ to both sides: $$x^2 + 4x + 4 = 9\$$
Factor the left side: $$ (x + 2)^2 = 9 \$$
Take the square root of both sides: $$ x + 2 = \pm 3 \$$
Solve for $$x\$$ : $$ x = -2 \pm 3 \$$
Final solutions: $$ x = 1 \$$ or $$ x = -5 \$$

Validation: Substitute $$x = 1\$$ and $$x = -5\$$ into the original equation:

$$ 2(1)^2 + 8(1) - 10 = 2 + 8 - 10 = 0 \$$
$$ 2(-5)^2 + 8(-5) - 10 = 50 - 40 - 10 = 0 \$$ ✓

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.