Solving Radical Equations Find The Value Of V In √v+7-4=3

Jul 7, 2025 by Admin 58 views

Solve for $v$, Where $v$ is a Real Number

Introduction

In this article, we will walk through the step-by-step solution for the equation $\sqrt{v+7}-4=3$ , where $v$ is a real number. This equation involves a square root, which means we need to isolate the square root term and then square both sides of the equation to eliminate the radical. It's important to check our solutions at the end to ensure they are valid, as squaring both sides can sometimes introduce extraneous solutions. Let's dive into the process and solve for $v$ methodically.

Step-by-Step Solution

To solve the equation $\sqrt{v+7}-4=3$ , we'll follow these steps:

Isolate the square root term: Our first goal is to get the square root term by itself on one side of the equation. We can do this by adding 4 to both sides of the equation:
$\sqrt{v+7}-4+4=3+4$ $\sqrt{v+7}=7$
Square both sides: To eliminate the square root, we square both sides of the equation:
$(\sqrt{v+7})^2=7^2$ $v+7=49$
Solve for $v$ : Now, we solve for $v$ by subtracting 7 from both sides of the equation:
$v+7-7=49-7$ $v=42$
Check the solution: It is crucial to check our solution in the original equation to make sure it is valid and not an extraneous solution. Plug $v=42$ back into the original equation:
$\sqrt{42+7}-4=3$ $\sqrt{49}-4=3$ $7-4=3$ $3=3$
Since the equation holds true, $v=42$ is a valid solution.

Detailed Explanation of Each Step

Let's delve deeper into each step to ensure a comprehensive understanding of the solution process.

Isolating the Square Root Term

The initial equation is $\sqrt{v+7}-4=3$ . To isolate the square root, we focus on getting the $\sqrt{v+7}$ term by itself. This is a fundamental step in solving equations involving radicals. By adding 4 to both sides, we maintain the equation's balance while moving the constant term to the right side. This results in the simplified equation $\sqrt{v+7}=7$ . Isolating the square root is crucial because it sets us up to eliminate the radical by squaring both sides, which is the next logical step in solving for $v$ . This isolation technique is a common strategy in algebra and is essential for handling more complex equations as well.

Squaring Both Sides

After isolating the square root term, the next step is to eliminate the square root. We achieve this by squaring both sides of the equation $\sqrt{v+7}=7$ . Squaring a square root effectively cancels out the radical, leaving us with the expression inside the square root. On the left side, $(\sqrt{v+7})^2$ simplifies to $v+7$ , and on the right side, $7^2$ equals 49. Thus, we obtain the equation $v+7=49$ . Squaring both sides is a powerful tool in solving equations with radicals, but it's also a step where extraneous solutions can be introduced. This is why checking the solution in the original equation is a critical final step. Understanding the principle of squaring to eliminate radicals is vital for solving various algebraic problems.

Solving for $v$

With the equation simplified to $v+7=49$ , solving for $v$ becomes a straightforward algebraic manipulation. Our goal is to isolate $v$ on one side of the equation. We achieve this by subtracting 7 from both sides, ensuring that the equation remains balanced. This operation gives us $v+7-7=49-7$ , which simplifies to $v=42$ . This step is a classic example of using inverse operations to solve for a variable. By subtracting 7, we effectively undo the addition of 7 to $v$ , allowing us to determine the value of $v$ . This method of isolating a variable is a core concept in algebra and is used extensively in solving linear equations and more complex mathematical problems.

Checking the Solution

Checking the solution is a crucial step, especially when dealing with equations involving square roots or other radicals. Squaring both sides of an equation can sometimes introduce extraneous solutions, which are values that satisfy the transformed equation but not the original one. To check our solution $v=42$ , we substitute it back into the original equation $\sqrt{v+7}-4=3$ . This gives us $\sqrt{42+7}-4=3$ , which simplifies to $\sqrt{49}-4=3$ . Further simplification yields $7-4=3$ , and finally, $3=3$ . Since this is a true statement, we confirm that $v=42$ is a valid solution. This verification step is not just a formality; it ensures the accuracy of our solution and demonstrates a thorough understanding of the problem-solving process.

Common Mistakes and How to Avoid Them

When solving equations involving square roots, several common mistakes can lead to incorrect solutions. Recognizing these pitfalls and understanding how to avoid them is crucial for success.

Forgetting to Check for Extraneous Solutions

One of the most common mistakes is failing to check the solution in the original equation. As mentioned earlier, squaring both sides of an equation can introduce extraneous solutions. These are values that satisfy the squared equation but not the original radical equation. For instance, if we had another equation and found a solution that, when checked, resulted in a negative value under the square root, that solution would be extraneous. To avoid this, always substitute your solution back into the original equation and verify that it holds true. This step is not optional; it's an essential part of the problem-solving process.

Incorrectly Squaring Both Sides

Another frequent error is incorrectly squaring both sides of the equation, especially when there are multiple terms. For example, if the equation were $\sqrt{v} + 2 = 5$ , some might mistakenly square each term individually, resulting in $v + 4 = 25$ , which is incorrect. The correct approach is to isolate the square root first, getting $\sqrt{v} = 3$ , and then square both sides, giving $v = 9$ . Similarly, if the equation involved $(\sqrt{v} + 2)^2$ , it's crucial to remember to expand the binomial correctly using the FOIL method or the binomial theorem. Accuracy in algebraic manipulation is key to avoiding these mistakes.

Misunderstanding the Domain of Square Roots

Square roots are only defined for non-negative numbers in the real number system. This means that the expression inside the square root (the radicand) must be greater than or equal to zero. Failing to consider this can lead to incorrect solutions. For instance, in the equation $\sqrt{v+7}$ , the expression $v+7$ must be non-negative. This implies that $v+7 \geq 0$ , which means $v \geq -7$ . If a solution is found that does not satisfy this condition, it is not a valid solution. Being mindful of the domain restrictions imposed by square roots is essential for accurate problem-solving.

Not Isolating the Square Root Term First

A common mistake is squaring both sides of the equation before isolating the square root term. This often leads to more complex equations that are harder to solve. For example, in the original equation $\sqrt{v+7}-4=3$ , if we squared both sides without isolating the square root first, we would get $(\sqrt{v+7}-4)^2=9$ . Expanding the left side would result in $v+7 - 8\sqrt{v+7} + 16 = 9$ , which is significantly more complicated to solve than the original equation. Always isolate the square root term before squaring both sides to simplify the process.

Arithmetic Errors

Simple arithmetic mistakes can also lead to incorrect solutions. These can occur during any step of the process, from adding or subtracting terms to squaring numbers. For example, a mistake in squaring 7, writing 48 instead of 49, would lead to an incorrect value for $v$ . To minimize these errors, it's helpful to double-check each step and perform calculations carefully. Writing out each step clearly and methodically can also help in identifying and correcting any arithmetic errors.

Conclusion

In summary, solving the equation $\sqrt{v+7}-4=3$ involves isolating the square root term, squaring both sides, solving for $v$ , and most importantly, checking the solution to avoid extraneous results. We found that $v=42$ is the valid solution for this equation. By understanding the step-by-step process and being mindful of common mistakes, you can confidently solve similar equations involving square roots. Remember to always verify your solutions to ensure accuracy and avoid extraneous results. This methodical approach not only helps in solving mathematical problems but also enhances your problem-solving skills in general.

Therefore, the solution to the equation $\sqrt{v+7}-4=3$ is:

v = 42