Solving the Equation (sqrt{x^2-8x 16} 2): A Comprehensive Guide

Often, we come across problems in algebra that require us to solve complex equations involving square roots and quadratic expressions. One such equation is based on the form (sqrt{x^2-8x 16} 2). This article provides a detailed walkthrough of the solution steps along with the mathematical background necessary to understand and solve such problems effectively.

Introduction to the Problem

Given the equation:

(sqrt{x^2-8x 16} 2)

Our goal is to find the values of (x) that satisfy this equation. Let's break down the problem step-by-step.

Square Both Sides

The first step involves squaring both sides of the equation to eliminate the square root. This is a common technique in algebra when dealing with square roots. By squaring both sides, the equation becomes:

(x^2 - 8x 16 4)

Manipulate to Form a Quadratic Equation

Once we have squared both sides, we can rearrange the equation to form a quadratic equation. Subtract 16 from both sides to isolate (x^2 - 8x) on the left-hand side:

(x^2 - 8x 4 - 16)

This simplifies to:

(x^2 - 8x -12)

Next, we will complete the square on the left-hand side. To do this, we need to add ((frac{-8}{2})^2 16) to both sides. This completes the square and transforms the left-hand side into a perfect square trinomial:

(x^2 - 8x 16 -12 16)

This simplifies to:

((x - 4)^2 4)

Solve for (x)

Finally, we take the square root of both sides to solve for (x). Remember that the square root of a number can be both positive and negative:

(x - 4 pm 2)

This gives us two possible solutions for (x):

(x - 4 2 Rightarrow x 6)

(x - 4 -2 Rightarrow x 2)

Therefore, the solutions to the equation (sqrt{x^2-8x 16} 2) are (x 6) and (x 2).

Conclusion

In summary, solving the equation (sqrt{x^2-8x 16} 2) involves several key steps: squaring both sides, simplifying the resulting quadratic equation, completing the square, and finally solving the square root equation by considering both positive and negative roots. These techniques are fundamental in algebra and form the basis for solving more complex polynomial and radical equations.

To further enhance your understanding of similar problems, you may explore additional resources on square root equations, quadratic equations, and algebraic manipulations. Practicing more examples will help solidify your understanding and improve problem-solving skills in algebra.