Solving The Equation Sqrt(x+2)-15=-3 A Step-by-Step Guide

In this article, we will delve into solving the algebraic equation $\sqrt{x+2}-15=-3$. This equation involves a square root, which means we need to isolate the radical term and then square both sides to eliminate the square root. It's a common type of problem encountered in algebra, and understanding the steps to solve it is crucial for building a solid foundation in mathematics. This type of equation often appears in various fields, including physics, engineering, and computer science, making it essential to master the techniques for solving them. We will explore each step meticulously, providing explanations and justifications to ensure clarity. By the end of this article, you will have a comprehensive understanding of how to solve this specific equation and similar equations involving square roots.

The equation $\sqrt{x+2}-15=-3$ presents a classic algebraic challenge involving a radical term. The square root symbol, denoted as $\sqrt{}$, signifies the principal (non-negative) square root of a number. Here, $\sqrt{x+2}$ represents the square root of the expression $x+2$. The equation states that when we take the square root of $x+2$ and then subtract 15, the result is -3. To solve for $x$, our goal is to isolate $x$ on one side of the equation. This involves several algebraic manipulations, including isolating the radical term and squaring both sides of the equation. Isolating the radical is a critical first step, as it allows us to eliminate the square root by squaring. We must also be mindful of extraneous solutions, which can arise when squaring both sides of an equation. Extraneous solutions are solutions that satisfy the transformed equation but not the original equation. Therefore, it's essential to check our solutions in the original equation to ensure they are valid. This step-by-step approach ensures that we not only find a solution but also verify its correctness.

Step 1: Isolate the Radical Term

The first step in solving the equation $\sqrt{x+2}-15=-3$ is to isolate the radical term, $\sqrt{x+2}$. This means we want to get the term with the square root by itself on one side of the equation. To do this, we add 15 to both sides of the equation:

$$\sqrt{x+2} - 15 + 15 = -3 + 15$$

This simplifies to:

$$\sqrt{x+2} = 12$$

Now, we have the square root term isolated on the left side, which sets us up for the next step of eliminating the square root. Isolating the radical is crucial because it allows us to directly address the square root by squaring both sides, which is the next logical step in solving this type of equation. By isolating the radical, we transform the equation into a form where we can easily eliminate the square root and solve for the variable $x$. This step is fundamental in solving any equation that involves radicals.

Step 2: Square Both Sides

Now that we have isolated the radical term, $\sqrt{x+2} = 12$, we can eliminate the square root by squaring both sides of the equation. Squaring both sides is a valid algebraic operation as long as we apply it to the entire expression on each side. This maintains the equality of the equation. Squaring $\sqrt{x+2}$ will undo the square root, leaving us with $x+2$. Squaring 12 will give us $12^2$, which is 144. So, we have:

$$(\sqrt{x+2})^2 = 12^2$$

This simplifies to:

$$x+2 = 144$$

By squaring both sides, we have transformed the equation from one involving a square root to a simple linear equation, which is much easier to solve. Squaring both sides is a key technique in dealing with radical equations, allowing us to eliminate the square root and proceed with solving for the variable.

Step 3: Solve for $x$

After squaring both sides, we now have the equation $x+2 = 144$. To solve for $x$, we need to isolate $x$ on one side of the equation. This can be done by subtracting 2 from both sides:

$$x + 2 - 2 = 144 - 2$$

This simplifies to:

$$x = 142$$

So, we have found a potential solution for $x$, which is 142. However, it's crucial to remember that squaring both sides of an equation can sometimes introduce extraneous solutions. Therefore, we need to verify this solution in the original equation to ensure it is valid. Solving for x involves basic algebraic manipulation, but the context of having squared both sides necessitates the verification step to avoid extraneous solutions. The solution $x=142$ is a candidate, but we must confirm it satisfies the original equation.

Step 4: Check the Solution

As mentioned earlier, squaring both sides of an equation can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Therefore, it is crucial to check our solution in the original equation, which is $\sqrt{x+2}-15=-3$. We found that $x = 142$, so we will substitute this value back into the original equation:

$$\sqrt{142+2}-15=-3$$

First, simplify inside the square root:

$$\sqrt{144}-15=-3$$

Now, find the square root of 144, which is 12:

$$12-15=-3$$

Finally, subtract 15 from 12:

$$-3 = -3$$

Since the equation holds true, $x = 142$ is a valid solution. Checking the solution is a critical step in solving radical equations, ensuring that we do not include any extraneous solutions in our final answer. This verification confirms that our solution is indeed correct and satisfies the original equation.

After completing the steps to isolate the radical, square both sides, solve for $x$, and check our solution, we have arrived at the final answer. The solution to the equation $\sqrt{x+2}-15=-3$ is $x = 142$. We verified this solution by substituting it back into the original equation and confirming that it satisfies the equation. This comprehensive process ensures that our answer is accurate and complete. The final answer, $x=142$, represents the value that makes the original equation true, and our verification step confirms its validity. This process highlights the importance of not only finding a solution but also confirming its correctness, especially when dealing with radical equations.

In this article, we have thoroughly explored the process of solving the equation $\sqrt{x+2}-15=-3$. We began by understanding the equation and the importance of isolating the radical term. We then meticulously followed the steps, which included isolating the square root, squaring both sides, solving for $x$, and most importantly, checking the solution to avoid extraneous results. The solution we found and verified is $x = 142$. The process of solving this equation demonstrates fundamental algebraic techniques applicable to a wide range of mathematical problems. Understanding and applying these techniques is crucial for success in algebra and beyond. This step-by-step approach not only helps in solving the equation but also builds a strong foundation for tackling more complex mathematical challenges in the future.