Solving The Radical Equation What Is The Solution Of $\sqrt{x-4}+5=2$

Decoding the Enigma: A Step-by-Step Solution

Let's embark on a mathematical journey to unravel the solution to the equation $\sqrt{x-4}+5=2$. This equation, adorned with a radical expression, presents a unique challenge that requires a systematic approach. Our quest begins with isolating the radical term, a crucial step in simplifying the equation. To achieve this, we subtract 5 from both sides, effectively disentangling the square root from its numerical companion. This transformation yields the equation $\sqrt{x-4} = -3$. Now, we encounter a pivotal moment in our exploration. The square root of any real number, by definition, cannot be negative. This fundamental principle casts a shadow of doubt on the existence of a solution. The left-hand side of the equation, $\sqrt{x-4}$, represents a non-negative value, while the right-hand side, -3, is a starkly negative number. This inherent contradiction signals a critical juncture in our analysis. It proclaims that no real number can satisfy this equation. The square root function, the cornerstone of our equation, is defined to produce non-negative outputs. Therefore, any attempt to equate it to a negative value is destined to fail. This inherent mathematical constraint renders the equation unsolvable within the realm of real numbers. The implication is profound: the solution set is empty. We can confidently assert that there is no real number 'x' that can make this equation hold true. The initial allure of finding a numerical answer fades as we confront the stark reality of mathematical impossibility. Our quest for a solution culminates in the resounding declaration of its non-existence. The equation, despite its seemingly straightforward appearance, harbors a subtle mathematical barrier that defies conventional solution-seeking methods. This exploration underscores the importance of understanding the underlying principles of mathematical functions. The square root function, with its inherent non-negativity, plays a pivotal role in shaping the solvability of equations. Ignoring this fundamental aspect can lead to futile attempts to find solutions where none exist. Our journey through this radical equation serves as a testament to the power of mathematical reasoning. It highlights the need to not only manipulate equations but also to critically assess the nature of the functions involved. The absence of a solution is not merely a negative outcome; it is a profound insight into the mathematical fabric of the equation itself. It compels us to appreciate the boundaries of mathematical possibility and the elegance of logical deduction.

The Illusion of Solutions: Why Extraneous Roots Lurk

In the realm of radical equations, the quest for solutions often involves squaring both sides, a seemingly innocuous maneuver that can inadvertently introduce extraneous roots – those deceptive numerical values that satisfy the transformed equation but fail to appease the original. These extraneous solutions, like mathematical mirages, can lead us astray if we are not vigilant in verifying our results. To illuminate this phenomenon, let's consider the equation $\sqrt{x-4}+5=2$ once more. As we discovered earlier, this equation has no real solution. However, let's pretend for a moment that we are unaware of this fact and proceed with the conventional method of squaring both sides. First, we isolate the radical term, obtaining $\sqrt{x-4} = -3$. Now, we square both sides, a step that transforms the equation into $(\sqrt{x-4})^2 = (-3)^2$, which simplifies to $x-4 = 9$. Solving for 'x', we arrive at $x = 13$. At first glance, this value might appear to be a legitimate solution. However, this is where the danger of extraneous roots lurks. We must subject this potential solution to a rigorous test, plugging it back into the original equation. Substituting $x = 13$ into $\sqrt{x-4}+5=2$, we get $\sqrt{13-4}+5=2$, which simplifies to $\sqrt{9}+5=2$, or $3+5=2$. This equation, alas, is a blatant falsehood. The left-hand side, 8, is nowhere near the right-hand side, 2. This glaring discrepancy exposes $x = 13$ as an imposter, an extraneous root that gained entry into our solution set through the act of squaring. The squaring operation, while a powerful tool for eliminating radicals, can also create new solutions that were not present in the original equation. This is because squaring can mask the sign of a number. For instance, both 3 and -3, when squared, yield 9. This ambiguity can lead to extraneous solutions if we are not careful to verify our answers. The moral of the story is clear: when dealing with radical equations, verification is paramount. Every potential solution must be scrutinized, subjected to the test of the original equation. Only those values that emerge unscathed from this trial can be deemed true solutions. The others, the extraneous roots, must be cast aside, banished from our solution set. Our exploration of extraneous roots highlights the importance of a nuanced understanding of mathematical operations. Squaring, while often a helpful technique, can also introduce complexities that demand careful attention. The vigilant solver must be aware of this potential pitfall and take appropriate measures to avoid it. The journey through the realm of radical equations is not merely about finding numerical answers; it is about navigating the subtle intricacies of mathematical transformations and the potential for deception. Extraneous roots serve as a reminder that mathematical solutions are not always what they seem. They demand a critical eye, a skeptical mind, and a commitment to rigorous verification.

The Domain's Dictate: Restricting the Realm of Possibilities

The domain of a function, like a gatekeeper, dictates the permissible inputs, the values that can be fed into the function without causing mathematical mayhem. In the realm of radical equations, the domain takes on a particularly crucial role, often serving as the first line of defense against nonsensical solutions. The square root function, the star of our equation $\sqrt{x-4}+5=2$, has a strict domain requirement: its argument, the expression under the radical sign, must be non-negative. This is because the square root of a negative number is not a real number. This domain restriction, seemingly a minor technicality, has profound implications for the solutions we can obtain. It narrows the playing field, limiting the values of 'x' that can even be considered as potential solutions. For our equation, the domain dictates that $x-4$ must be greater than or equal to zero. This inequality, a simple yet powerful statement, unveils the permissible range for 'x'. Adding 4 to both sides, we discover that $x \geq 4$. This inequality, the domain's decree, informs us that 'x' must be at least 4. Any value less than 4 is forbidden, for it would lead to the square root of a negative number, a mathematical transgression. Now, let's revisit our earlier attempt to solve the equation. We arrived at the potential solution $x = 13$. This value, as we verified, is an extraneous root. However, had we considered the domain restriction beforehand, we might have spared ourselves the trouble of plugging it back into the original equation. The domain's dictate, $x \geq 4$, does not, in itself, disqualify $x = 13$. It merely sets the stage for a valid solution. But the fact that $x = 13$ ultimately failed to satisfy the original equation underscores the importance of domain awareness. The domain acts as a filter, weeding out values that are fundamentally incompatible with the square root function. It helps us to avoid pursuing solutions that are destined to be extraneous. In the case of our equation, the domain, while not immediately revealing the absence of a solution, provides a crucial context for understanding the equation's behavior. It reminds us that the square root function is not defined for all real numbers. It compels us to consider the potential limitations imposed by the radical expression. The domain's role extends beyond merely identifying extraneous solutions. It also helps us to understand the nature of the solution set. If the domain is empty, then the equation is guaranteed to have no solutions. If the domain is restricted, then we know that any solutions must lie within those boundaries. The domain, in essence, provides a framework for our solution-seeking endeavors. It guides our steps, preventing us from straying into mathematical wilderness. It is a silent sentinel, guarding against nonsensical results and ensuring that our solutions are grounded in mathematical reality. Our exploration of the domain highlights the importance of a holistic approach to equation solving. It is not enough to simply manipulate symbols; we must also understand the underlying functions and their limitations. The domain, the often-overlooked aspect of mathematical functions, plays a vital role in shaping the landscape of solutions.

The Verdict: No Escape from the Absence of a Solution

After our detailed exploration, the verdict is clear: the equation $\sqrt{x-4}+5=2$ has no solution. This conclusion, reached through a combination of algebraic manipulation and careful consideration of the square root function's properties, underscores the importance of a rigorous approach to equation solving. We began by attempting to isolate the radical term, a standard technique for tackling equations involving square roots. This led us to the equation $\sqrt{x-4} = -3$. At this juncture, we encountered a fundamental contradiction. The square root of any real number, by definition, cannot be negative. This inherent mathematical constraint rendered the equation unsolvable within the realm of real numbers. We delved into the concept of extraneous roots, those deceptive solutions that emerge from the act of squaring both sides of an equation. We discovered that the potential solution $x = 13$, obtained by squaring, was in fact an extraneous root, failing to satisfy the original equation. This exploration highlighted the importance of verification, of plugging potential solutions back into the original equation to ensure their legitimacy. We also examined the role of the domain, the set of permissible inputs for a function. The domain restriction for the square root function, namely that its argument must be non-negative, further reinforced the absence of a solution. The domain, in this case, did not directly reveal the lack of a solution, but it provided a crucial context for understanding the equation's behavior. The absence of a solution is not merely a negative outcome; it is a profound insight into the mathematical fabric of the equation itself. It reveals a fundamental incompatibility between the terms involved, a barrier that cannot be overcome through algebraic manipulation. Our journey through this equation serves as a testament to the power of mathematical reasoning. It highlights the need to not only manipulate equations but also to critically assess the nature of the functions involved. The square root function, with its inherent non-negativity, plays a pivotal role in shaping the solvability of equations. Ignoring this fundamental aspect can lead to futile attempts to find solutions where none exist. The equation $\sqrt{x-4}+5=2$, despite its seemingly straightforward appearance, harbors a subtle mathematical barrier that defies conventional solution-seeking methods. This exploration underscores the importance of understanding the underlying principles of mathematical functions. The absence of a solution is not a sign of failure; it is a testament to the rigor of mathematical logic. It is a reminder that not all equations have solutions, and that the quest for solutions must be guided by a deep understanding of mathematical principles. In conclusion, the equation $\sqrt{x-4}+5=2$ stands as a compelling example of a mathematical enigma with no solution. Its resolution requires a blend of algebraic skill, awareness of function properties, and a commitment to rigorous verification. The journey through this equation, while ultimately leading to a null result, offers valuable lessons in mathematical reasoning and the art of problem-solving.

Selecting the Correct Answer: The Definitive Choice

Based on our comprehensive analysis, the correct answer is D. no solution. The equation $\sqrt{x-4}+5=2$ has no real solution due to the inherent contradiction arising from the non-negativity of the square root function. Any attempt to solve the equation algebraically leads to a false statement, confirming the absence of a solution. This conclusion is further reinforced by the domain restriction of the square root function, which dictates that the expression under the radical sign must be non-negative. The other options, A. $x=-17$, B. $x=13$, and C. $x=53$, are all incorrect. These values, when substituted into the original equation, do not yield a true statement. Option B, $x=13$, represents an extraneous root, a value that satisfies a transformed version of the equation but not the original. The selection of option D, no solution, is the culmination of our mathematical exploration. It is the definitive answer, the only choice that accurately reflects the equation's unsolvability.