Solving The Radical Equation √(-3l - 6) = 5 A Step By Step Guide
In this article, we will delve into the process of solving the equation √(-3l - 6) = 5, ensuring a clear and comprehensive understanding of each step involved. Our primary goal is to find the value(s) of 'l' that satisfy the given equation. We'll explore the fundamental concepts of algebraic manipulation and radical equations, providing a step-by-step solution while emphasizing the importance of verifying our results. If no real solution exists, we will confidently declare "DNE" (Does Not Exist). If multiple solutions emerge, we will meticulously separate them with commas. Understanding how to tackle such equations is crucial for anyone studying algebra and beyond. This comprehensive guide aims to make the solution process accessible and straightforward, whether you're a student tackling homework problems or simply looking to refresh your algebra skills. Let's embark on this mathematical journey together and unlock the solution to the equation √(-3l - 6) = 5.
Understanding the Basics of Radical Equations
Before diving into the specific steps to solve the equation √(-3l - 6) = 5, it’s crucial to grasp the fundamental concepts of radical equations. A radical equation is simply an equation where the variable is located inside a radical, most commonly a square root. Solving these equations requires a systematic approach to eliminate the radical and isolate the variable. The most common technique involves squaring both sides of the equation, but this step can introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original. Therefore, verification is a crucial step in solving radical equations. The underlying principle relies on the property that if a = b, then a² = b², but the reverse isn't always true. Understanding the properties of square roots and the potential for extraneous solutions is paramount in accurately solving these types of equations. When dealing with square roots, we are primarily concerned with the principal (non-negative) square root. This means that the expression under the square root (the radicand) must be non-negative for the solution to be real. This constraint plays a significant role in determining the existence of real solutions for the equation √(-3l - 6) = 5. The process of squaring both sides allows us to transform the radical equation into a more manageable form, typically a linear or quadratic equation, which we can then solve using standard algebraic techniques. However, the potential for extraneous solutions necessitates careful verification, ensuring that the solutions obtained are valid in the context of the original equation. Understanding these fundamental concepts will enable us to approach the given equation with confidence and accuracy.
Step-by-Step Solution for √(-3l - 6) = 5
Now, let's meticulously solve the equation √(-3l - 6) = 5 using a step-by-step approach. This ensures clarity and accuracy in our solution. Remember, the goal is to isolate the variable 'l'.
Step 1: Squaring Both Sides
The first critical step in solving this radical equation is to eliminate the square root. We achieve this by squaring both sides of the equation. This operation is based on the principle that if two quantities are equal, their squares are also equal. Squaring both sides of the equation √(-3l - 6) = 5 gives us:
(√(-3l - 6))² = 5²
This simplifies to:
-3l - 6 = 25
By squaring both sides, we've successfully transformed the equation from a radical equation into a simple linear equation, which is much easier to solve.
Step 2: Isolating the Variable 'l'
Having eliminated the square root, our next objective is to isolate the variable 'l'. This involves performing algebraic operations to get 'l' by itself on one side of the equation. We start by adding 6 to both sides of the equation -3l - 6 = 25:
-3l - 6 + 6 = 25 + 6
This simplifies to:
-3l = 31
Next, we divide both sides of the equation by -3 to solve for 'l':
-3l / -3 = 31 / -3
This gives us:
l = -31/3
At this point, we have a potential solution for 'l'. However, it is crucial to remember that squaring both sides of an equation can introduce extraneous solutions, so the next step is essential.
Step 3: Verifying the Solution
Verification is a paramount step in solving radical equations. We need to ensure that our potential solution, l = -31/3, satisfies the original equation √(-3l - 6) = 5. We substitute l = -31/3 back into the original equation:
√(-3 * (-31/3) - 6) = 5
This simplifies to:
√(31 - 6) = 5
√25 = 5
5 = 5
Since the equation holds true, our solution l = -31/3 is valid.
The Final Solution
After a methodical step-by-step solution and rigorous verification, we have confidently arrived at the solution for the equation √(-3l - 6) = 5. The solution is:
l = -31/3
This solution is a reduced fraction, fulfilling the requirements of the problem. We have successfully navigated through the process of solving a radical equation, highlighting the importance of squaring both sides, isolating the variable, and most importantly, verifying the solution. This comprehensive approach ensures that we obtain accurate results and avoid extraneous solutions.
Common Mistakes and How to Avoid Them
Solving radical equations like √(-3l - 6) = 5 can be tricky, and it’s easy to make mistakes if you’re not careful. Let's discuss some common pitfalls and how to avoid them.
Forgetting to Verify Solutions
One of the most frequent errors is neglecting to verify the solutions obtained after squaring both sides of the equation. This step is absolutely crucial because squaring can introduce extraneous solutions, which are values that satisfy the transformed equation but not the original one. To avoid this, always substitute your potential solutions back into the original equation to check if they hold true. In our example, we correctly verified that l = -31/3 is a valid solution by plugging it back into √(-3l - 6) = 5.
Incorrectly Squaring Both Sides
Another common mistake is making errors when squaring both sides of the equation. It’s essential to square the entire side, not just individual terms. For instance, if you had an equation like √(a + b) = c, you should square the entire left side to get (√(a + b))² = (a + b), and the right side to get c². Ensure you are squaring both sides completely and correctly to maintain the equation's balance.
Arithmetic Errors
Simple arithmetic mistakes can lead to incorrect solutions. This includes errors in addition, subtraction, multiplication, and division, especially when dealing with fractions and negative numbers. Double-checking your calculations at each step can help prevent these errors. For example, when solving -3l = 31, be cautious when dividing by -3 to ensure you get the correct sign for l.
Ignoring Domain Restrictions
When dealing with square roots, it’s crucial to consider the domain restrictions. The expression under the square root (the radicand) must be non-negative. Failing to account for this can lead to incorrect conclusions about the existence of real solutions. Before solving, check the radicand to see if there are any restrictions on the variable. In our equation √(-3l - 6) = 5, we implicitly ensured that -3l - 6 ≥ 0 for the square root to be real.
Misunderstanding the Definition of Square Root
Remember that the square root symbol (√) denotes the principal (non-negative) square root. This means that √25 = 5, not ±5. This understanding is crucial when verifying solutions and ensuring they fit the original equation. By being mindful of these common pitfalls and taking the necessary precautions, you can confidently and accurately solve radical equations.
Conclusion
In conclusion, solving the equation √(-3l - 6) = 5 involves a systematic approach that includes squaring both sides, isolating the variable, and crucially, verifying the solution. We found that l = -31/3 is the valid solution to this equation. Throughout this article, we've emphasized the importance of understanding the underlying principles of radical equations and the potential pitfalls, such as extraneous solutions. By following the step-by-step methods and being mindful of common mistakes, you can confidently tackle similar algebraic challenges. Remember, practice and attention to detail are key to mastering these concepts. Whether you're a student or simply looking to sharpen your math skills, a thorough understanding of these techniques will serve you well. This comprehensive guide aims to equip you with the knowledge and confidence to approach and solve radical equations effectively.
