Roots Of The Equation $\sqrt{x^2-5x+5}-1=0$ Solving And Analysis

Jul 13, 2025 by ADMIN 65 views

$Unveiling the Roots of the Equation: A Deep Dive into $\sqrt{x^2-5 x+5}-1=0$$

In the realm of mathematics, equations serve as the bedrock of problem-solving and critical thinking. Among the diverse types of equations, those involving square roots often present a unique set of challenges and require meticulous analysis. This article delves into the intricacies of solving the equation $\sqrt{x^2-5 x+5}-1=0$ , aiming to determine the nature of its roots and identify the correct statement among the given options.

Deconstructing the Equation: A Step-by-Step Approach

To unravel the mysteries of this equation, we embark on a systematic journey, employing algebraic techniques to isolate the variable and unveil its potential values. Our first step involves isolating the square root term, paving the way for further simplification.

Isolating the Square Root:

We begin by adding 1 to both sides of the equation, effectively isolating the square root term on one side:

$\sqrt{x^2-5 x+5} = 1$

Eliminating the Square Root:

To eliminate the square root, we square both sides of the equation. This operation is crucial in removing the radical sign and transforming the equation into a more manageable form. However, it's essential to acknowledge that squaring both sides can sometimes introduce extraneous solutions, which are values that satisfy the transformed equation but not the original one. Therefore, we must verify our solutions later to ensure their validity.

$(\sqrt{x^2-5 x+5})^2 = 1^2$

$x^2-5 x+5 = 1$

Rearranging into a Quadratic Equation:

Our next step involves rearranging the equation into the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$ . This form allows us to readily apply established methods for solving quadratic equations.

Subtracting 1 from both sides, we obtain:

$x^2-5 x+4 = 0$

Solving the Quadratic Equation:

Now, we have a classic quadratic equation that we can solve using various methods, such as factoring, completing the square, or employing the quadratic formula. In this case, factoring proves to be the most efficient approach.

We seek two numbers that multiply to 4 and add up to -5. These numbers are -1 and -4. Thus, we can factor the quadratic equation as follows:

$(x-1)(x-4) = 0$

Identifying Potential Roots:

The factored form of the equation reveals the potential roots. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, either $(x-1) = 0$ or $(x-4) = 0$ .

Solving for $x$ in each case, we find two potential roots:

$x = 1$
$x = 4$

Verifying the Roots: Separating Valid Solutions from Extraneous Ones

As we previously noted, squaring both sides of an equation can introduce extraneous solutions. Therefore, it's imperative to verify our potential roots by substituting them back into the original equation. This step ensures that the solutions we obtain are valid and not artifacts of the transformation process.

Verifying x = 1:

Substituting $x = 1$ into the original equation, we get:

$\sqrt{1^2-5(1)+5}-1 = 0$

$\sqrt{1-5+5}-1 = 0$

$\sqrt{1}-1 = 0$

$1-1 = 0$

$0 = 0$

The equation holds true for $x = 1$ , confirming that it is a valid root.

Verifying x = 4:

Substituting $x = 4$ into the original equation, we get:

$\sqrt{4^2-5(4)+5}-1 = 0$

$\sqrt{16-20+5}-1 = 0$

$\sqrt{1}-1 = 0$

$1-1 = 0$

$0 = 0$

The equation also holds true for $x = 4$ , indicating that it is another valid root.

Conclusion: Unveiling the True Nature of the Roots

Through our meticulous step-by-step analysis, we have successfully identified the roots of the equation $\sqrt{x^2-5 x+5}-1=0$ . We discovered that both $x = 1$ and $x = 4$ satisfy the original equation, making them valid solutions.

Therefore, the correct statement is:

c) Both 1 and 4 are roots.

This exploration highlights the importance of not only solving equations but also verifying the solutions obtained. By diligently checking for extraneous roots, we can ensure the accuracy and validity of our results. This approach is particularly crucial when dealing with equations involving square roots or other transformations that may introduce extraneous solutions.

Delving Deeper: The Significance of Root Verification

Root verification is not merely a formality; it is an integral part of the problem-solving process, especially when dealing with equations that have undergone transformations. Squaring both sides of an equation, while a common technique to eliminate square roots, can inadvertently introduce solutions that do not satisfy the original equation. These extraneous solutions arise because the squaring operation can mask the original sign of the expression. For instance, both 2 and -2, when squared, result in 4. Therefore, squaring both sides can lead to solutions that satisfy the squared equation but not the original equation involving the square root.

Consider the equation $\sqrt{x} = -2$ . If we square both sides, we get $x = 4$ . However, substituting $x = 4$ back into the original equation yields $\sqrt{4} = -2$ , which simplifies to $2 = -2$ , a clear contradiction. This example vividly illustrates the importance of verifying solutions to weed out extraneous roots.

The process of verification involves substituting each potential solution back into the original equation and checking if the equation holds true. If the substitution leads to a true statement, the solution is valid. If the substitution results in a contradiction, the solution is extraneous and must be discarded. This meticulous verification process ensures that only the true roots of the equation are retained.

Beyond the Equation: The Broader Implications of Mathematical Rigor

The exercise of solving and verifying the roots of the equation $\sqrt{x^2-5 x+5}-1=0$ extends beyond the specific problem itself. It underscores the importance of mathematical rigor and the need for a systematic approach to problem-solving. Each step in the process, from isolating the square root to solving the quadratic equation and verifying the roots, demands precision and attention to detail. This methodical approach not only leads to the correct solution but also cultivates critical thinking skills that are valuable in various domains.

The process of solving this equation exemplifies the power of algebraic manipulation and the elegance of mathematical reasoning. It demonstrates how seemingly complex equations can be解构into simpler components and solved using established techniques. Moreover, the emphasis on verification highlights the importance of questioning assumptions and ensuring the validity of solutions.

In the broader context of mathematics, the concepts explored in this article are fundamental building blocks. Understanding how to solve equations involving square roots and other transformations is essential for tackling more advanced mathematical problems. The principles of root verification and the avoidance of extraneous solutions are applicable to a wide range of mathematical contexts, including calculus, differential equations, and complex analysis.

Mastering the Art of Equation Solving: A Path to Mathematical Proficiency

Solving equations is a cornerstone of mathematics, and mastering this skill is crucial for achieving proficiency in the subject. The equation $\sqrt{x^2-5 x+5}-1=0$ provides a valuable opportunity to hone equation-solving techniques and develop a deeper understanding of mathematical principles.

By following a systematic approach, we can confidently tackle a wide range of equations, regardless of their complexity. The key lies in breaking down the problem into smaller, manageable steps, applying appropriate algebraic techniques, and verifying the solutions obtained. This methodical approach not only ensures accuracy but also fosters a sense of accomplishment and confidence in one's mathematical abilities.

The journey of solving equations is not just about finding the right answer; it's about developing critical thinking skills, fostering logical reasoning, and appreciating the elegance of mathematical structures. As we delve deeper into the world of equations, we unlock new dimensions of mathematical understanding and pave the way for further exploration and discovery.

In conclusion, the equation $\sqrt{x^2-5 x+5}-1=0$ serves as a compelling example of the importance of careful analysis, verification, and mathematical rigor. By understanding the nuances of equation solving and the potential pitfalls of extraneous solutions, we can navigate the complexities of mathematics with confidence and precision.