Solve Quadratic Equations Using Square Roots - Algebra-2

1. Fundamental Concepts

Definition: A quadratic equation is an equation of the form $$ax^2 + bx + c = 0\$$ , where $$a \neq 0\$$ .
Square Root Method: This method is used to solve quadratic equations of the form $$x^2 = k\$$ , where $$k\$$ is a constant.
Principal Square Root: The principal square root of a number $$k\$$ is the non-negative value that, when squared, gives $$k\$$ .

2. Key Concepts

Square Root Property: If $$x^2 = k\$$ , then $$x = \pm \sqrt{k}\$$

Isolation of $$x^2\$$ : To use the square root method, isolate $$x^2\$$ 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 $$x^2 = 16\$$

Step-by-Step Solution:

Apply the square root property: $$x = \pm \sqrt{16}\$$
Simplify: $$x = \pm 4\$$

Validation: Substitute $$x = 4\$$ and $$x = -4\$$ into the original equation: $$4^2 = 16\$$ and $$(-4)^2 = 16\$$ ✓

Example 2 (Intermediate)

Problem: Solve $$2x^2 - 18 = 0\$$

Step-by-Step Solution:

Isolate $$x^2\$$ : $$2x^2 = 18\$$
Divide both sides by 2: $$x^2 = 9\$$
Apply the square root property: $$x = \pm \sqrt{9}\$$
Simplify: $$x = \pm 3\$$

Validation: Substitute $$x = 3\$$ and $$x = -3\$$ into the original equation: $$2(3)^2 - 18 = 0\$$ and $$2(-3)^2 - 18 = 0\$$ ✓

4. Problem-Solving Techniques

Isolation Technique: Always isolate $$x^2\$$ 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.