Solving The Equation √{x+2} = 4-x A Comprehensive Guide

In this article, we delve into the process of solving the equation ${\sqrt{x+2} = 4-x}$. This type of equation, which involves a square root, requires careful steps to ensure we arrive at the correct solution. We will explore each step in detail, explain the reasoning behind it, and ultimately determine the interval to which the solution belongs. This exploration not only helps in understanding the specific problem but also provides a broader insight into handling radical equations. Throughout this guide, we will emphasize the importance of checking for extraneous solutions, a common pitfall in radical equations. By the end of this article, you will have a clear understanding of how to solve this equation and similar problems, enhancing your mathematical toolkit.

Understanding the Problem

The given equation is ${\sqrt{x+2} = 4-x}$. To find the solution, we need to isolate ${x}$. The first step involves eliminating the square root. This is crucial because the presence of the square root complicates the algebraic manipulations. By understanding this initial step, we lay the groundwork for a systematic solution. The equation represents a relationship between a square root function and a linear function. Solving it means finding the ${x}$ value(s) where these two functions intersect. This gives us a geometric perspective on the problem, which can be helpful in visualizing the solution. Additionally, recognizing the domain of the square root function is essential. Since we cannot take the square root of a negative number in the real number system, we must ensure that ${x+2 \geq 0}$, which implies ${x \geq -2}$. This constraint is vital in checking for extraneous solutions later on.

Step-by-Step Solution

Eliminating the Square Root

To eliminate the square root in the equation ${\sqrt{x+2} = 4-x}$, we square both sides. Squaring both sides is a standard technique for dealing with radical equations. This gives us:

${(\sqrt{x+2})^2 = (4-x)^2}$

This simplifies to:

${x+2 = (4-x)(4-x)}$

Expanding and Simplifying

Next, we expand the right side of the equation:

${x+2 = 16 - 8x + x^2}$

Rearranging the terms to form a quadratic equation, we get:

${x^2 - 9x + 14 = 0}$

This step is crucial as it transforms the original radical equation into a more manageable quadratic equation. Quadratic equations are well-studied, and we have several methods to solve them, such as factoring, completing the square, or using the quadratic formula. The key here is to ensure that the rearrangement is done accurately, as any mistake in this step will lead to incorrect solutions. The quadratic equation represents a parabola, and its roots are the ${x}$-intercepts of the parabola. Finding these roots is equivalent to solving the quadratic equation.

Solving the Quadratic Equation

We can solve the quadratic equation ${x^2 - 9x + 14 = 0}$ by factoring. We look for two numbers that multiply to 14 and add to -9. These numbers are -2 and -7. Thus, we can factor the quadratic equation as:

${(x-2)(x-7) = 0}$

Setting each factor equal to zero gives us the potential solutions:

${x-2 = 0 \Rightarrow x = 2}$

${x-7 = 0 \Rightarrow x = 7}$

So, we have two potential solutions: ${x = 2}$ and ${x = 7}$. Factoring is an efficient method for solving quadratic equations when the roots are integers or simple fractions. However, it is not always possible to factor a quadratic equation, in which case we would need to resort to other methods like the quadratic formula. The solutions we have found are only potential solutions at this stage. Because we squared both sides of the original equation, we introduced the possibility of extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. This is why the next step is crucial.

Checking for Extraneous Solutions

It is essential to check these potential solutions in the original equation ${\sqrt{x+2} = 4-x}$ to identify any extraneous solutions.

Checking ${x = 2}$

Substitute ${x = 2}$ into the original equation:

${\sqrt{2+2} = 4-2}$

${\sqrt{4} = 2}$

${2 = 2}$

This is true, so ${x = 2}$ is a valid solution.

Checking ${x = 7}$

Substitute ${x = 7}$ into the original equation:

${\sqrt{7+2} = 4-7}$

${\sqrt{9} = -3}$

${3 = -3}$

This is false, so ${x = 7}$ is an extraneous solution. Extraneous solutions arise because squaring both sides of an equation can introduce solutions that do not satisfy the original equation. The process of squaring can change the fundamental nature of the equation, leading to these false solutions. Therefore, checking solutions in the original equation is not just a formality, but a critical step in solving radical equations. The check for ${x=7}$ clearly shows that it does not satisfy the original equation, highlighting the importance of this step.

The Valid Solution

Therefore, the only valid solution is ${x = 2}$. This emphasizes the importance of checking for extraneous solutions when dealing with radical equations. The process of squaring both sides can introduce solutions that do not actually satisfy the original equation, making it crucial to verify each potential solution. In this case, only one of the potential solutions, ${x = 2}$, holds true when substituted back into the original equation. This careful approach ensures that we arrive at the correct answer and avoid including any false solutions.

Determining the Interval

Now we need to determine which interval the solution ${x = 2}$ belongs to:

a) ${12, 7}$ b) ${2, 3}$ c) ${0, 1}$ d) ${-1, 3}$ e) ${-1, 1}$

The solution ${x = 2}$ belongs to the interval ${2, 3) and ${[-1, 3]}$. To identify the correct interval, we simply check which of the given intervals contains the value 2. Interval a) [12, 7}$ is invalid as intervals are written from the smallest to the largest number. Interval b) ${2, 3}$ includes all numbers between 2 and 3, including 2. Interval c) ${0, 1}$ includes numbers between 0 and 1, which does not include 2. Interval d) ${-1, 3}$ includes numbers between -1 and 3, which includes 2. Interval e) ${-1, 1}$ includes numbers between -1 and 1, which does not include 2. Thus, both intervals b) and d) contain the solution ${x = 2}$. However, since we are looking for the most specific interval, we choose the smallest one that contains the solution. This step is a straightforward application of understanding interval notation and how to check if a number belongs to a given interval. The ability to correctly identify the interval demonstrates a solid understanding of number ranges and their representation.

Final Answer

The solution ${x = 2}$ belongs to the interval [2, 3] and [-1, 3]. This concludes the process of solving the equation and identifying the correct interval. The journey from the initial equation to the final answer involved several critical steps, each requiring careful attention to detail. From eliminating the square root to solving the quadratic equation, checking for extraneous solutions, and finally, identifying the correct interval, every step played a crucial role in ensuring the accuracy of the final result. This comprehensive approach not only solves the problem at hand but also reinforces the importance of a methodical approach to mathematical problem-solving. The process highlights the interconnectedness of various mathematical concepts and techniques, making it a valuable learning experience.

Conclusion

In summary, solving the equation ${\sqrt{x+2} = 4-x}$ involves squaring both sides, simplifying to a quadratic equation, solving the quadratic equation, and crucially, checking for extraneous solutions. The valid solution, ${x = 2}$, belongs to the interval [2, 3] and [-1, 3]. This exercise demonstrates the importance of a systematic approach to solving radical equations, the necessity of verifying solutions, and the application of various algebraic techniques. Understanding these steps is vital for mastering not just this specific type of problem, but also for tackling a broader range of mathematical challenges. The process underscores the value of precision, attention to detail, and a thorough understanding of underlying mathematical principles. By following these guidelines, one can confidently approach and solve similar problems, enhancing their problem-solving skills and mathematical proficiency.