Solving The Equation Sqrt(x^2+49) = X+5

ADMIN · · 11 min read

Table Of Content

  1. Introduction
  2. Detailed Solution
  3. Conceptual Understanding
    1. Importance of Checking for Extraneous Solutions
    2. The Domain of Square Root Functions
    3. Alternative Methods and Considerations
    4. Common Mistakes to Avoid
  4. Conclusion

Introduction

In this article, we will delve into solving the equation $\sqrt{x^2+49}=x+5$. This equation involves a square root, which requires careful algebraic manipulation to isolate the variable x and find its possible values. We will explore the steps involved in solving this equation, including squaring both sides, simplifying the resulting quadratic equation, and checking for extraneous solutions. Understanding how to solve such equations is crucial in various fields of mathematics, physics, and engineering. Let's begin by outlining the detailed solution and then discussing the underlying concepts and potential pitfalls.

Detailed Solution

To solve the equation $\sqrt{x^2+49}=x+5$, we'll follow these steps:

  1. Isolate the Square Root: The square root term is already isolated on the left side of the equation, which simplifies our initial step.

  2. Square Both Sides: To eliminate the square root, we square both sides of the equation:

    $(\sqrt{x2+49})2 = (x+5)^2$

    This simplifies to:

    $x^2 + 49 = x^2 + 10x + 25$

  3. Simplify the Equation: Subtract $x^2$ from both sides:

    $49 = 10x + 25$

    Now, subtract 25 from both sides:

    $24 = 10x$

  4. Solve for x: Divide both sides by 10:

    $x = \frac{24}{10} = \frac{12}{5}$

  5. Check for Extraneous Solutions: Since we squared both sides of the equation, it's essential to check our solution in the original equation. Substitute $x = \frac{12}{5}$ into the original equation:

    $\sqrt{(\frac{12}{5})^2+49} = \frac{12}{5}+5$

    $\sqrt{\frac{144}{25}+49} = \frac{12}{5}+5$

    $\sqrt{\frac{144}{25}+\frac{1225}{25}} = \frac{12}{5}+\frac{25}{5}$

    $\sqrt{\frac{1369}{25}} = \frac{37}{5}$

    $\frac{37}{5} = \frac{37}{5}$

    The solution checks out.

Conceptual Understanding

Importance of Checking for Extraneous Solutions

When solving equations involving radicals (like square roots), it's crucial to check for extraneous solutions. Extraneous solutions are values that satisfy the transformed equation (after squaring in this case) but do not satisfy the original equation. This happens because squaring both sides of an equation can introduce solutions that were not present initially. For example, consider the equation $\sqrt{x} = -2$. Squaring both sides gives $x = 4$, but substituting $x = 4$ back into the original equation yields $\sqrt{4} = 2$, which is not equal to -2. Hence, $x = 4$ is an extraneous solution.

The Domain of Square Root Functions

The domain of a square root function is the set of all x-values for which the expression inside the square root is non-negative. In our original equation, $\sqrt{x^2+49}$, the expression inside the square root, $x^2+49$, is always positive for any real number x because $x^2$ is always non-negative, and adding 49 ensures it remains positive. Therefore, there are no domain restrictions from the square root in this particular problem. However, it's a vital consideration in general when dealing with square root equations.

Alternative Methods and Considerations

While squaring both sides is a common method for solving equations with square roots, it's not the only approach. In some cases, one might use substitution or graphical methods, especially when dealing with more complex equations or systems of equations. Graphically, solving $\sqrt{x^2+49}=x+5$ involves finding the intersection points of the graphs $y = \sqrt{x^2+49}$ and $y = x+5$. The x-coordinate of the intersection point(s) represents the solution(s) to the equation.

Common Mistakes to Avoid

  1. Forgetting to Check for Extraneous Solutions: This is a critical step, and omitting it can lead to incorrect answers.

  2. Incorrectly Squaring Binomials: When squaring $(x+5)$, remember to use the correct formula $(a+b)^2 = a^2 + 2ab + b^2$, avoiding common errors like $(x+5)^2 = x^2 + 25$.

  3. Misunderstanding the Domain: Always consider the domain of the square root function to ensure the solutions are valid.

  4. Algebraic Errors: Errors in algebraic manipulation, such as incorrectly simplifying terms, can lead to wrong solutions.

Conclusion

The solution to the equation $\sqrt{x^2+49}=x+5$ is $x = \frac{12}{5}$. By squaring both sides, simplifying, and checking for extraneous solutions, we arrived at this result. This process illustrates the importance of careful algebraic manipulation and the necessity of verifying solutions in radical equations. Understanding these concepts and techniques is fundamental for solving more complex mathematical problems. Remember to always check for extraneous solutions and consider the domain restrictions when dealing with square root functions.

Related Posts