Solving the Equation x-22 16: Steps and Solutions

When faced with the equation x-22 16, the objective is to find the values of x that satisfy this equation. This process involves understanding the properties of square roots and solving the resulting linear and quadratic equations. Let's break down the steps in detail to understand how to approach and solve this equation.

Step 1: Understanding the Problem

The given equation is x-22 16. To make it clearer, we can rewrite the equation as:

x - 22 16

Step 2: Isolating the Square Term

To isolate the square term, let's move the constant term on the right-hand side to the left:

x - 4 16

Adding 4 to both sides, we get:

x 16 4

x 20

However, this step is not fully addressing the original equation. Let's revisit the initial form and take the square root of both sides to solve for x.

Step 3: Taking the Square Root of Both Sides

Starting again from the original equation:

x - 22 16

We take the square root of both sides:

sqrt[x - 22] sqrt[16]

This simplifies to:

x - 2 4 or x - 2 -4

Step 4: Solving Each Linear Equation

We now set up and solve each of the linear equations resulting from the square root operation.

1. Solving x - 2 4

x - 2 4

A. Add 2 to both sides:

x 4 2

B. Simplify:

x 6

2. Solving x - 2 -4

x - 2 -4

A. Add 2 to both sides:

x -4 2

B. Simplify:

x -2

Final Solutions

Therefore, the solutions to the equation x - 22 16 are:

x 6 and x -2

Key Concepts to Remember

1. Square Root and Absolute Value: Taking the square root introduces both positive and negative solutions, and the absolute value of a number squared is the number itself, irrespective of whether it is positive or negative.

2. Solving Quadratic Equations: The square root of a number squared results in the absolute value of that number, which is why we get two possible solutions.

3. Checking Extraneous Solutions: It's important to double-check each solution by plugging it back into the original equation to ensure it's valid.

Conclusion

In summary, the solutions to the equation x - 22 16 are x 6 and x -2. Understanding the properties of square roots and quadratic equations is crucial in solving such problems effectively. Remembering these key concepts will help you tackle similar algebraic equations with confidence.