Finding The Quadratic Equation With Solutions X = (5 ± 2√7) / 3
To determine which quadratic equation has the solutions x = (5 ± 2√7) / 3, we will work backward from the solutions to construct the equation. This involves understanding the relationship between the roots of a quadratic equation and its coefficients. This exploration will not only provide the answer but also delve into the mathematical principles that govern quadratic equations and their solutions. Understanding how to derive a quadratic equation from its solutions is a fundamental skill in algebra, with applications extending to various fields such as physics, engineering, and computer science. This article aims to provide a comprehensive explanation, ensuring a solid grasp of the underlying concepts and techniques. We'll also touch upon common pitfalls and strategies for verifying your solutions, making this a valuable resource for students and educators alike. The journey from solutions to equation involves reversing the steps typically used to solve quadratic equations, offering a unique perspective on the subject matter. By the end of this discussion, you should be well-equipped to tackle similar problems and appreciate the elegance and interconnectedness of mathematical concepts. Moreover, this process highlights the importance of algebraic manipulation and the power of working backward to solve problems, skills that are crucial in advanced mathematical studies.
Understanding the Quadratic Formula and Roots
Before diving into the specifics, let's recap the quadratic formula. For a quadratic equation in the standard form ax² + bx + c = 0, the solutions for x are given by the quadratic formula:
x = (-b ± √(b² - 4ac)) / (2a)
In our case, the solutions are x = (5 ± 2√7) / 3. We need to manipulate this form to match it with the quadratic formula. Identifying the components of the quadratic formula within the given solutions is crucial. The term outside the square root corresponds to -b/(2a), and the term inside the square root, after some manipulation, will relate to b² - 4ac. This step-by-step deconstruction is essential for reconstructing the quadratic equation. The quadratic formula is a cornerstone of algebra, and understanding its derivation and application is paramount. It not only provides a method for solving quadratic equations but also reveals the nature of the roots based on the discriminant (b² - 4ac). The discriminant determines whether the roots are real and distinct, real and equal, or complex. This understanding allows for a deeper analysis of quadratic equations and their behavior. Moreover, the quadratic formula has historical significance, tracing back to ancient civilizations that grappled with solving quadratic problems. Its enduring relevance underscores its importance in mathematics and its applications in various scientific and engineering disciplines. By mastering the quadratic formula, one gains a powerful tool for solving a wide range of mathematical problems.
Reconstructing the Quadratic Equation
Given x = (5 ± 2√7) / 3, we can infer that the denominator 3 corresponds to 2a in the quadratic formula. Thus, 2a = 3, and a can be considered as a multiple of 3/2 or simply we can try a = 3 by multiplying both sides. Let's rewrite the solutions to better align with the quadratic formula structure. Multiplying both sides of the equation x = (5 ± 2√7) / 3 by 3, we get 3x = 5 ± 2√7. This manipulation brings us closer to identifying the b and c coefficients of the quadratic equation. The goal here is to reverse the steps taken when applying the quadratic formula, effectively undoing the operations. This process highlights the symmetry inherent in mathematical operations and the importance of understanding inverse relationships. The act of reconstructing the equation also reinforces the connection between the roots and the coefficients, a fundamental concept in algebra. Furthermore, this method provides a valuable problem-solving strategy: working backward from the solution to find the original problem. This approach can be applied to various mathematical contexts and is a testament to the versatility of algebraic thinking. By carefully manipulating the equation and paying attention to the structure of the quadratic formula, we can systematically derive the corresponding quadratic equation. This exercise not only enhances our understanding of quadratic equations but also strengthens our algebraic skills.
Now, let's isolate the square root term: 3x - 5 = ± 2√7. Squaring both sides will eliminate the square root: (3x - 5)² = (± 2√7)². Expanding the left side, we get (3x - 5)² = 9x² - 30x + 25. On the right side, (± 2√7)² = 4 * 7 = 28. Thus, we have the equation 9x² - 30x + 25 = 28. Subtracting 28 from both sides gives us 9x² - 30x - 3 = 0. Now, we can simplify the equation by dividing all terms by their greatest common divisor, which is 3. This yields 3x² - 10x - 1 = 0.
This resulting equation is in the standard quadratic form, and it directly corresponds to one of the options provided. The process of squaring both sides is a common technique in algebra, but it's crucial to remember that it can sometimes introduce extraneous solutions. However, in this case, because we are working backward from known solutions, we are not concerned with extraneous solutions. The step of expanding the squared term requires careful attention to detail, ensuring that all terms are correctly multiplied and combined. The simplification step, dividing by the greatest common divisor, is essential for presenting the equation in its simplest form and making it easier to compare with the given options. This entire process underscores the importance of algebraic manipulation and the ability to transform equations while preserving their solutions. By carefully applying these techniques, we can effectively reverse-engineer quadratic equations from their roots, demonstrating a deep understanding of the underlying mathematical principles.
Matching with the Given Options
Comparing our derived equation, 3x² - 10x - 1 = 0, with the options provided, we find that it matches option 4.
- 3x² – 5x + 7 = 0
- 3x² – 5x – 1 = 0
- 3x² – 10x + 6 = 0 4) 3x² – 10x – 1 = 0
Therefore, the quadratic equation with solutions x = (5 ± 2√7) / 3 is 3x² – 10x – 1 = 0. This verification step is crucial in any mathematical problem-solving process. It ensures that the derived solution aligns with the given options and confirms the accuracy of the steps taken. In this case, the direct match provides strong confidence in our result. However, it's always a good practice to double-check the calculations and reasoning to eliminate any potential errors. This meticulous approach is essential for building trust in one's mathematical abilities and for preventing mistakes in more complex problems. Furthermore, the act of matching the derived equation with the options reinforces the understanding of the relationship between the equation and its solutions. It highlights the fact that there is a unique quadratic equation associated with a given pair of solutions (in this case, a conjugate pair). This understanding is fundamental to solving quadratic equations and related problems. By carefully comparing the derived equation with the options, we not only find the correct answer but also solidify our grasp of the underlying mathematical concepts.
Alternative Method: Sum and Product of Roots
Another method to find the quadratic equation involves using the sum and product of the roots. If x₁ and x₂ are the roots of the quadratic equation ax² + bx + c = 0, then:
- Sum of roots: x₁ + x₂ = -b/a
- Product of roots: x₁ * x₂ = c/a
In our case, x₁ = (5 + 2√7) / 3 and x₂ = (5 - 2√7) / 3. First, let's find the sum of the roots:
x₁ + x₂ = ((5 + 2√7) / 3) + ((5 - 2√7) / 3) = (5 + 2√7 + 5 - 2√7) / 3 = 10 / 3
Next, let's find the product of the roots:
x₁ * x₂ = ((5 + 2√7) / 3) * ((5 - 2√7) / 3) = (25 - (4 * 7)) / 9 = (25 - 28) / 9 = -3 / 9 = -1 / 3
Now, consider the quadratic equation in the form x² - (sum of roots)x + (product of roots) = 0. Substituting the values we found:
x² - (10 / 3)x + (-1 / 3) = 0
To eliminate the fractions, multiply the entire equation by 3:
3x² - 10x - 1 = 0
This method provides an alternative approach to finding the quadratic equation and reinforces the connection between the roots and the coefficients. The sum and product of roots are fundamental properties of quadratic equations and offer a powerful tool for solving related problems. This approach is particularly useful when the roots are known, as it provides a direct pathway to constructing the equation. Understanding the relationship between the sum and product of roots and the coefficients of the quadratic equation is crucial for a deeper understanding of quadratic theory. This method also highlights the elegance and interconnectedness of mathematical concepts, demonstrating how different approaches can lead to the same solution. Furthermore, this alternative method serves as a valuable check on our previous result, increasing our confidence in the accuracy of our solution. By employing multiple methods, we can gain a more comprehensive understanding of the problem and develop stronger problem-solving skills.
Conclusion
We have successfully determined that the quadratic equation with solutions x = (5 ± 2√7) / 3 is 3x² – 10x – 1 = 0, which corresponds to option 4. We achieved this through two methods: first, by reversing the quadratic formula, and second, by using the sum and product of roots. Both methods provide valuable insights into the nature of quadratic equations and their solutions. This exploration underscores the importance of understanding the fundamental principles of algebra and the versatility of mathematical problem-solving techniques. The ability to work backward from solutions to equations is a crucial skill in various mathematical contexts and highlights the interconnectedness of mathematical concepts. Moreover, the use of multiple methods not only provides a means of verification but also deepens our understanding of the problem at hand. The quadratic formula and the sum and product of roots are powerful tools in algebra, and mastering them is essential for success in higher-level mathematics. This discussion serves as a valuable resource for students and educators alike, providing a comprehensive explanation of how to find a quadratic equation given its solutions. By understanding these methods, one can confidently tackle similar problems and appreciate the beauty and elegance of mathematical reasoning.
