Solving Quadratic Equations Using The Quadratic Formula Finding The Solution Set
The quadratic formula stands as a cornerstone in algebra, providing a universal method for solving quadratic equations of the form ax² + bx + c = 0. This formula is particularly invaluable when factoring proves challenging or impossible. In this article, we will delve into a specific scenario where the quadratic formula yields a solution of the form (2 ± √36) / 6, and our objective is to meticulously determine the solution set for the equation. Understanding the nuances of the quadratic formula and its application is crucial for students and anyone involved in mathematical problem-solving. We will explore each step in detail, ensuring clarity and comprehension. By the end of this discussion, you will have a solid grasp of how to interpret such results and identify the correct solution set.
Understanding the Quadratic Formula and Its Components
At its core, the quadratic formula is a powerful tool designed to find the roots, or solutions, of any quadratic equation structured as ax² + bx + c = 0. These roots represent the x-intercepts of the parabola defined by the quadratic equation. The formula itself is expressed as x = (-b ± √(b² - 4ac)) / (2a). Here, 'a,' 'b,' and 'c' are coefficients derived directly from the quadratic equation, where 'a' is the coefficient of the x² term, 'b' is the coefficient of the x term, and 'c' is the constant term. A critical component of the formula is the discriminant, represented by b² - 4ac. The discriminant offers vital information about the nature of the roots. If the discriminant is positive, the quadratic equation has two distinct real roots. A discriminant of zero indicates exactly one real root (a repeated root), and a negative discriminant implies two complex roots. The ± symbol in the quadratic formula signifies that there are generally two solutions, one obtained by adding the square root term and the other by subtracting it. This dual solution aspect is a direct consequence of the parabolic nature of quadratic functions, which typically intersect the x-axis at two points unless the vertex of the parabola lies on the x-axis (one real root) or the parabola does not intersect the x-axis at all (complex roots). Mastering the quadratic formula involves not only memorizing the formula itself but also understanding how to correctly identify and substitute the coefficients a, b, and c, and how to interpret the discriminant to anticipate the nature of the solutions. This comprehensive understanding is essential for accurately solving a wide range of quadratic equations and for further applications in algebra and beyond.
Deciphering the Given Solution: (2 ± √36) / 6
The provided solution, (2 ± √36) / 6, is a direct outcome of applying the quadratic formula to a specific quadratic equation. To fully understand this solution, we must break it down and simplify each component. The expression consists of two potential solutions, one derived from addition and the other from subtraction, due to the ± symbol. The square root term, √36, is a perfect square, simplifying to 6. This simplification is crucial for obtaining the final, concise solutions. We can now rewrite the expression as (2 ± 6) / 6. This means we have two separate calculations to perform: (2 + 6) / 6 and (2 - 6) / 6. The first calculation, (2 + 6) / 6, results in 8 / 6, which can be further simplified to 4 / 3. The second calculation, (2 - 6) / 6, yields -4 / 6, which simplifies to -2 / 3. These two simplified fractions, 4 / 3 and -2 / 3, represent the two distinct roots of the quadratic equation. They are the x-values where the parabola intersects the x-axis. Understanding how to simplify such expressions is a fundamental skill in algebra, and it is essential for accurately determining the solution set of any quadratic equation. The ability to recognize perfect squares within square roots and to reduce fractions to their simplest forms is particularly important. In this case, the clear and step-by-step simplification process allows us to confidently identify the two solutions and proceed to determine the correct solution set from the given options.
Identifying the Correct Solution Set
Now that we have simplified the solution (2 ± √36) / 6 to its component roots, 4 / 3 and -2 / 3, our next step is to accurately identify the solution set. The solution set is a collection of all values that satisfy the original quadratic equation, and in this case, it comprises the two roots we have calculated. The solutions we found are x = 4 / 3 and x = -2 / 3. The solution set is typically represented using set notation, where the elements are enclosed in curly braces and separated by commas. Therefore, the solution set for this equation is {-2 / 3, 4 / 3}. We must now carefully compare this result with the given options to determine the correct answer. The options are presented in various formats, and it's essential to recognize the correct representation of the solution set. Option A, (-2 / 3, 4 / 3), represents an open interval, which is not the correct format for a solution set. Open intervals denote all real numbers between the specified endpoints, excluding the endpoints themselves, which is not what we are looking for. Option B, (2), suggests a single solution, which contradicts our finding of two distinct roots. Option C, (1, 3), presents two different values that do not match our calculated roots. This discrepancy confirms that Option C is also incorrect. By comparing our calculated solution set {-2 / 3, 4 / 3} with the given options, we can confidently identify the correct solution set. It is crucial to understand the notation used for solution sets, intervals, and individual solutions to avoid confusion and ensure accurate selection. This step-by-step comparison process is essential for verifying the correctness of the solution and for demonstrating a thorough understanding of the problem-solving process.
Conclusion: The Solution Set and Its Implications
In conclusion, by applying the quadratic formula and simplifying the resulting expression (2 ± √36) / 6, we have successfully determined the solution set for the quadratic equation in question. The simplification process yielded two distinct roots: -2 / 3 and 4 / 3. These roots, when correctly represented in set notation, form the solution set {-2 / 3, 4 / 3}. Through careful comparison with the provided options, we identified that this solution set corresponds to option A. This exercise highlights the importance of a thorough understanding of the quadratic formula, the simplification of radical expressions, and the correct interpretation of solution sets. The ability to accurately solve quadratic equations is a fundamental skill in algebra and has wide-ranging applications in various fields of mathematics and science. Moreover, the process of breaking down a complex problem into smaller, manageable steps, as demonstrated in this analysis, is a valuable problem-solving strategy applicable to a wide range of scenarios. Understanding the nature of quadratic equations, their solutions, and the various ways these solutions can be represented is crucial for mathematical literacy and for tackling more advanced mathematical concepts. This example serves as a clear illustration of how the quadratic formula can be used to solve real-world problems and reinforces the importance of attention to detail and accuracy in mathematical calculations.