Equivalent Equations Solving $x^2 - 10x - 11 = 0$

#h1 Select the Correct Answer to Determine Equivalent Equations

In this comprehensive guide, we will dissect the problem of identifying which equation shares the same solutions as the quadratic equation $x^2 - 10x - 11 = 0$. This task involves a blend of algebraic manipulation and a solid grasp of quadratic equation properties. We'll systematically explore each option, providing a step-by-step analysis to ensure clarity and understanding. Our focus will be on employing techniques like completing the square to transform the given equation and compare it with the provided choices. By the end of this discussion, you'll not only have the correct answer but also a deeper insight into the methodologies for solving such problems.

Understanding the Base Equation: $x^2 - 10x - 11 = 0$

Our starting point is the quadratic equation $x^2 - 10x - 11 = 0$. To find an equivalent equation, we need to manipulate this into a form that matches one of the options provided. A powerful technique for this is completing the square. This method allows us to rewrite the quadratic expression in a way that reveals the solutions more clearly. The process involves transforming the equation into the form $(x - h)^2 = k$, where $h$ and $k$ are constants. By doing so, we can directly compare our transformed equation with the given options and determine the correct match. This approach not only helps in solving the immediate problem but also enhances our understanding of quadratic equations and their properties.

To begin, let's isolate the terms involving $x$: $x^2 - 10x = 11$. Now, we need to add a constant to both sides to complete the square on the left-hand side. To find this constant, we take half of the coefficient of the $x$ term (which is -10), square it, and add it to both sides. Half of -10 is -5, and (-5)^2 is 25. Thus, we add 25 to both sides of the equation. This step is crucial because it allows us to rewrite the left side as a perfect square, simplifying the equation and bringing us closer to the desired form for comparison with the given options. This method showcases the elegance of algebraic manipulation in solving quadratic equations.

Adding 25 to both sides gives us $x^2 - 10x + 25 = 11 + 25$, which simplifies to $x^2 - 10x + 25 = 36$. The left-hand side is now a perfect square trinomial, which can be factored as $(x - 5)^2$. Thus, our equation becomes $(x - 5)^2 = 36$. This form is particularly useful as it directly relates to the squared terms in the options, making it easier to identify the correct answer. This transformation highlights the power of completing the square in simplifying and solving quadratic equations.

Analyzing the Options

Now that we have transformed the original equation into $(x - 5)^2 = 36$, we can compare it with the options provided to determine which one matches. This comparison is straightforward since our transformed equation is in a similar format to the options. Each option represents a quadratic equation in the form of a squared term equal to a constant, making the comparison quite direct. The key is to look for the option that has the same squared term and the same constant value on the right side of the equation. This step-by-step comparison is essential to ensure accuracy and to understand why one option is correct while the others are not.

Let's examine each option:

A. $(x - 10)^2 = 36$ B. $(x - 5)^2 = 36$ C. $(x - 10)^2 = 21$ D. $(x - 5)^2 = 21$

By comparing these options with our transformed equation $(x - 5)^2 = 36$, we can clearly see that option B is the correct match. Both the squared term $(x - 5)^2$ and the constant 36 are identical, confirming that this equation has the same solutions as the original equation. This direct comparison underscores the importance of transforming the equation into a standard form for easy analysis.

Options A, C, and D differ from our transformed equation in either the squared term or the constant value. Option A has a different squared term $(x - 10)^2$, while options C and D have different constant values (21 instead of 36). These differences indicate that these equations will have different solutions compared to the original equation. This detailed analysis of each option reinforces the understanding of how variations in the equation affect the solutions.

The Correct Answer: Option B

After a detailed analysis and comparison, we can confidently conclude that option B, $(x - 5)^2 = 36$, is the correct answer. This equation has the same solutions as the given equation $x^2 - 10x - 11 = 0$. Our method of completing the square allowed us to transform the original equation into a form that is directly comparable to the given options, making the identification of the correct answer straightforward. This methodical approach highlights the power of algebraic manipulation in solving mathematical problems. The solution not only provides the correct answer but also reinforces the understanding of the underlying mathematical principles.

Alternative Method: Solving for x

Another way to confirm our answer is by solving for $x$ in both the original equation and the selected option. This approach provides a direct verification that the solutions are indeed the same. By finding the roots of both equations, we can definitively prove their equivalence. This method is particularly useful for double-checking the solution and ensuring accuracy. It also offers a different perspective on the problem, reinforcing the understanding of how solutions are related to the equation's form.

First, let's solve the original equation $x^2 - 10x - 11 = 0$. We can factor this quadratic equation as $(x - 11)(x + 1) = 0$. Setting each factor equal to zero gives us the solutions $x = 11$ and $x = -1$. These values are the roots of the original equation and serve as a benchmark for comparison with the solutions of the selected option. This step is crucial in verifying the correctness of our initial solution.

Now, let's solve option B, $(x - 5)^2 = 36$. Taking the square root of both sides gives us $x - 5 = ±6$. This results in two equations: $x - 5 = 6$ and $x - 5 = -6$. Solving these equations gives us $x = 11$ and $x = -1$, which are the same solutions as the original equation. This direct verification confirms that option B is indeed the correct answer. The consistency in solutions reinforces the validity of our method and the accuracy of our result.

This alternative method of solving for $x$ provides a robust confirmation of our initial answer. By finding the roots of both equations and verifying their equivalence, we gain further confidence in our solution. This approach also highlights the interconnectedness of different algebraic techniques in solving quadratic equations.

Why Other Options Are Incorrect

To further solidify our understanding, let's briefly discuss why the other options are incorrect. This analysis will reinforce the concepts of equation equivalence and the importance of accurate algebraic manipulation. By understanding why certain options are wrong, we can better appreciate the nuances of solving quadratic equations and avoid common mistakes.

Option A, $(x - 10)^2 = 36$, is incorrect because the squared term $(x - 10)^2$ is different from the $(x - 5)^2$ term we derived from the original equation. This difference indicates that the vertex of the parabola represented by this equation is shifted compared to the original, leading to different solutions. Understanding the graphical representation of quadratic equations can provide further insight into why these differences matter.

Options C, $(x - 10)^2 = 21$, and D, $(x - 5)^2 = 21$, are incorrect because the constant term on the right side of the equation is different from 36. Changing the constant term affects the vertical position of the parabola and, consequently, the solutions of the equation. This highlights the sensitivity of quadratic equation solutions to changes in the constant term.

By recognizing these differences and understanding their implications, we can avoid common pitfalls in solving quadratic equations. This detailed analysis of incorrect options reinforces the importance of precision and accuracy in algebraic manipulation.

Conclusion

In conclusion, after a thorough step-by-step analysis, we have determined that option B, $(x - 5)^2 = 36$, is the equation with the same solutions as the given equation $x^2 - 10x - 11 = 0$. We arrived at this solution by completing the square, transforming the original equation into a comparable form, and verifying our answer through an alternative method of solving for $x$. This comprehensive approach not only provides the correct answer but also enhances our understanding of quadratic equations and their properties. The process underscores the importance of algebraic manipulation, equation equivalence, and the verification of solutions in mathematics. By mastering these techniques, we can confidently tackle a wide range of mathematical problems.