Solving √(-u-4) - √(u+5) = -1 A Step-by-Step Guide
In this comprehensive guide, we will delve into the step-by-step solution of the equation √(-u-4) - √(u+5) = -1, where u is a real number. This equation involves square roots and requires careful algebraic manipulation to isolate the variable u and determine its value. Our discussion will cover the initial setup, the algebraic steps to eliminate the square roots, verification of the solutions, and an insightful exploration of the underlying mathematical principles. This detailed explanation aims to provide a clear and thorough understanding of the problem-solving process, making it accessible to both students and math enthusiasts. Understanding the nuances of solving such equations is crucial for mastering algebra and related mathematical disciplines. This article will serve as a valuable resource, offering clarity and precision in each step, ensuring that readers can confidently tackle similar problems in the future.
Before we embark on solving the equation, it is imperative to understand the constraints imposed by the square roots. The expressions inside the square roots, -u-4 and u+5, must be non-negative to yield real number solutions. This gives us two inequalities:
- -u-4 ≥ 0
- u+5 ≥ 0
Let's solve these inequalities:
- For -u-4 ≥ 0, we add u to both sides to get -4 ≥ u, which can be rewritten as u ≤ -4.
- For u+5 ≥ 0, we subtract 5 from both sides to get u ≥ -5.
Combining these two inequalities, we find that the domain of u is -5 ≤ u ≤ -4. This means that any solution we find for u must lie within this interval. This preliminary step is crucial because it helps us avoid extraneous solutions, which can arise when dealing with square root equations. By establishing the domain at the outset, we ensure that our final solutions are mathematically valid. Ignoring this step can lead to incorrect results, making the initial setup a vital part of the problem-solving process. This rigorous approach not only aids in solving the current problem but also reinforces the importance of considering domain restrictions in all mathematical problems involving radicals.
Now, let's proceed with the algebraic manipulation to solve the equation √(-u-4) - √(u+5) = -1. Our primary goal is to eliminate the square roots to simplify the equation and isolate u. The first step is to isolate one of the square root terms. We can add √(u+5) to both sides of the equation:
√(-u-4) = √(u+5) - 1
Next, we square both sides of the equation to eliminate one of the square roots. Squaring both sides gives us:
(-u-4) = (√(u+5) - 1)²
Expanding the right side, we get:
-u-4 = (u+5) - 2√(u+5) + 1
Now, we simplify the equation by combining like terms:
-u-4 = u+6 - 2√(u+5)
To further isolate the remaining square root, we move all terms without a square root to one side of the equation:
2√(u+5) = 2u + 10
We can divide both sides by 2 to simplify:
√(u+5) = u + 5
Now, we square both sides again to eliminate the last square root:
(u+5) = (u+5)²
Expanding the right side, we have:
u+5 = u² + 10u + 25
Rearranging the terms to form a quadratic equation, we get:
u² + 9u + 20 = 0
This quadratic equation is now ready to be solved. The process of algebraic manipulation we've undertaken is crucial in transforming the original equation into a solvable form. Each step, from isolating square roots to squaring both sides, is meticulously performed to ensure accuracy and to gradually simplify the equation. This methodical approach not only helps in finding the correct solutions but also reinforces the importance of careful algebraic techniques in mathematical problem-solving.
We have now arrived at the quadratic equation u² + 9u + 20 = 0. To solve for u, we can use factoring, the quadratic formula, or completing the square. In this case, factoring is the most straightforward method. We are looking for two numbers that multiply to 20 and add up to 9. These numbers are 4 and 5. Thus, we can factor the quadratic equation as follows:
(u+4)(u+5) = 0
Setting each factor equal to zero gives us two possible solutions for u:
- u+4 = 0 → u = -4
- u+5 = 0 → u = -5
So, we have two potential solutions: u = -4 and u = -5. It is essential to remember that these are just potential solutions. We must verify them in the original equation to ensure they are not extraneous solutions. Extraneous solutions can arise when dealing with radical equations because the squaring process can introduce solutions that do not satisfy the original equation. This step of solving the quadratic equation is a critical juncture in our problem-solving process. It demonstrates the power of factoring in simplifying and solving polynomial equations. However, the journey is not complete until we confirm the validity of these solutions within the context of the original problem. This meticulous approach ensures mathematical rigor and accuracy in our final answer.
Now that we have found two potential solutions, u = -4 and u = -5, it is crucial to verify them in the original equation √(-u-4) - √(u+5) = -1 to ensure they are valid and not extraneous. Extraneous solutions can occur due to the squaring of both sides of the equation, which may introduce solutions that do not satisfy the original equation.
Let's verify u = -4:
√(-(-4)-4) - √((-4)+5) = √0 - √1 = 0 - 1 = -1
So, u = -4 satisfies the original equation.
Now, let's verify u = -5:
√(-(-5)-4) - √((-5)+5) = √1 - √0 = 1 - 0 = 1
Since 1 ≠ -1, u = -5 does not satisfy the original equation. Therefore, u = -5 is an extraneous solution.
After verification, we find that only u = -4 is a valid solution. This step of verification is paramount in solving equations involving radicals. It underscores the importance of not only finding potential solutions but also confirming their validity within the original problem context. The process of verifying solutions is a testament to the rigorous nature of mathematical problem-solving, ensuring that our final answer is both accurate and mathematically sound. This meticulous approach helps in avoiding errors and reinforces a deeper understanding of the solution process.
In conclusion, the equation √(-u-4) - √(u+5) = -1 has one valid solution: u = -4. We arrived at this solution through a series of steps, starting with understanding the domain restrictions imposed by the square roots, followed by algebraic manipulation to eliminate the radicals, solving the resulting quadratic equation, and finally, verifying the solutions in the original equation. The initial setup involved recognizing that -u-4 ≥ 0 and u+5 ≥ 0, which led us to the domain -5 ≤ u ≤ -4. The algebraic manipulation included isolating a square root, squaring both sides, and simplifying the equation to eventually obtain a quadratic equation. Solving the quadratic equation u² + 9u + 20 = 0 gave us two potential solutions: u = -4 and u = -5. However, it was the verification step that revealed u = -5 as an extraneous solution, leaving u = -4 as the only valid solution.
This comprehensive solution not only answers the question but also highlights the importance of each step in the problem-solving process. From understanding domain restrictions to verifying solutions, each step plays a crucial role in ensuring the accuracy and validity of the final answer. This methodical approach is essential in mathematics and provides a framework for tackling similar problems with confidence. The journey from the initial equation to the final solution underscores the beauty and precision of mathematics, where each step builds upon the previous one to reveal the answer. This detailed exploration serves as a valuable learning tool, reinforcing mathematical concepts and problem-solving strategies.
