Finding R In Quadratic Equation Solution (11 ± √r) / 2

In the realm of mathematics, quadratic equations stand as fundamental expressions, often taking the form ax² + bx + c = 0. Solving these equations is a cornerstone of algebraic understanding, and the quadratic formula serves as a powerful tool for this purpose. This article delves into a specific quadratic equation, x² - 11x + 5 = 0, and explores how its solutions, expressed in the form (11 ± √r) / 2, reveal a fascinating interplay between the coefficients and the discriminant. Our primary focus is to determine the value of 'r', a task that requires a careful application of the quadratic formula and a keen eye for algebraic manipulation. Understanding the components of the quadratic formula, particularly the discriminant, is key to unlocking the value of 'r'. The discriminant, represented as b² - 4ac, lies at the heart of determining the nature and values of the roots of a quadratic equation. It dictates whether the equation has two distinct real roots, one repeated real root, or two complex roots. By carefully comparing the given solution form with the general quadratic formula, we can effectively isolate the discriminant and solve for 'r'. This exploration not only strengthens our understanding of quadratic equations but also highlights the elegant connections within mathematical expressions. Moreover, this problem serves as an excellent example of how abstract algebraic concepts translate into concrete numerical values, fostering a deeper appreciation for the power and beauty of mathematics. So, let's embark on this mathematical journey and uncover the hidden value of 'r' within this intriguing quadratic equation. The process will involve identifying the coefficients, applying the quadratic formula, and then meticulously comparing the result with the given solution form. This methodical approach will not only lead us to the answer but also reinforce the importance of careful algebraic manipulation and attention to detail in mathematical problem-solving. Let's begin by identifying the coefficients of the given quadratic equation, a crucial first step in our quest to find the value of 'r'.

H2: The Quadratic Formula: A Key to Unlocking Solutions

The quadratic formula is the cornerstone of solving quadratic equations of the form ax² + bx + c = 0. This powerful formula provides a direct method for finding the roots (or solutions) of any quadratic equation, regardless of the complexity of its coefficients. The formula is expressed as: x = (-b ± √(b² - 4ac)) / 2a. Each term in the formula plays a critical role in determining the nature and value of the roots. The coefficients a, b, and c are derived directly from the quadratic equation itself, where a is the coefficient of the x² term, b is the coefficient of the x term, and c is the constant term. The expression b² - 4ac under the square root is known as the discriminant. The discriminant is a crucial component of the quadratic formula as it reveals the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it's zero, the equation has one repeated real root. And if it's negative, the equation has two complex roots. Understanding the discriminant is essential for predicting the type of solutions we can expect from a quadratic equation. In our specific problem, the solutions are given in a form that directly relates to the quadratic formula, which makes it an excellent opportunity to apply our knowledge of the formula. By comparing the given solution form with the general quadratic formula, we can pinpoint the value of the discriminant and subsequently determine the value of 'r'. The ± symbol in the quadratic formula signifies that there are generally two solutions to a quadratic equation, corresponding to the two possible signs of the square root term. These two solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis. The quadratic formula is not just a formula to memorize; it's a powerful tool derived from the process of completing the square, a fundamental algebraic technique. The elegance of the quadratic formula lies in its ability to provide a universal solution for any quadratic equation, regardless of the specific values of its coefficients. Before we can apply the quadratic formula to our problem, we need to correctly identify the coefficients a, b, and c from the given equation. This is a straightforward process but requires careful attention to the signs and values of each term. Once we have these coefficients, we can substitute them into the quadratic formula and simplify the expression to find the solutions. This will then allow us to compare our result with the given solution form and ultimately determine the value of 'r'. Let's now turn our attention to the specific quadratic equation in question and identify these key coefficients.

H2: Identifying Coefficients: The First Step

To effectively apply the quadratic formula, the first crucial step is to correctly identify the coefficients a, b, and c from the given quadratic equation. In our case, the equation is x² - 11x + 5 = 0. By comparing this equation with the standard form ax² + bx + c = 0, we can easily extract the values of these coefficients. The coefficient a is the number multiplying the x² term. In our equation, the x² term has an implied coefficient of 1, meaning a = 1. This is a common occurrence, especially when the quadratic equation is presented in a simplified form. The coefficient b is the number multiplying the x term. In our equation, the x term is -11x, so b = -11. It's crucial to pay close attention to the sign of the coefficient, as it directly affects the outcome of the quadratic formula. A mistake in the sign can lead to an incorrect solution. The coefficient c is the constant term, which is the term that does not involve x. In our equation, the constant term is 5, so c = 5. Identifying the constant term is usually straightforward, but it's important to ensure that it's included in the equation with the correct sign. Once we have correctly identified the coefficients a = 1, b = -11, and c = 5, we have laid the foundation for applying the quadratic formula. These values will be directly substituted into the formula, and the subsequent calculations will lead us to the solutions of the equation. The accuracy of our coefficient identification is paramount, as any error at this stage will propagate through the rest of the solution process. Therefore, it's always a good practice to double-check the values of a, b, and c before proceeding further. Now that we have confidently identified the coefficients, we are ready to plug them into the quadratic formula and begin the process of solving for x. This will involve careful substitution, simplification, and algebraic manipulation to arrive at the solutions in the desired form. Let's proceed with the substitution and witness how the quadratic formula transforms these coefficients into the roots of the equation. This step is where the power of the formula truly shines, allowing us to unravel the solutions hidden within the equation.

H2: Applying the Formula: Substituting and Simplifying

With the coefficients identified as a = 1, b = -11, and c = 5, the next step is to substitute these values into the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. This substitution is a critical juncture in the problem-solving process, as it directly translates the abstract coefficients into a concrete expression that can be simplified. Let's perform the substitution carefully. Replacing b with -11, we get -(-11) in the numerator, which simplifies to 11. Next, we substitute the values into the discriminant, b² - 4ac. This becomes (-11)² - 4(1)(5). Simplifying further, we get 121 - 20, which equals 101. Now, let's substitute a = 1 into the denominator, 2a, which becomes 2(1) = 2. Putting it all together, the quadratic formula now gives us: x = (11 ± √101) / 2. This expression represents the two solutions to the quadratic equation. The ± sign indicates that there are two possible values for x: one where we add the square root and one where we subtract it. The simplification process involves careful attention to the order of operations and the rules of arithmetic. Squaring the negative number -11 results in a positive value, 121. Multiplying 4, 1, and 5 gives us 20, which is then subtracted from 121. The square root of 101 cannot be simplified further, as 101 is a prime number. The denominator is simply 2 times 1, which is 2. This entire process demonstrates the power and elegance of the quadratic formula. By systematically substituting the coefficients and simplifying the expression, we have arrived at the solutions to the equation in a clear and concise form. Now, we have the solutions in the form x = (11 ± √101) / 2. The final step is to compare this with the given solution form, (11 ± √r) / 2, and determine the value of r. This comparison is the key to unlocking the answer to our problem. Let's proceed with this final comparison and unveil the value of 'r'. This step requires a keen eye for detail and a direct comparison of the terms in the two expressions.

H2: The Final Comparison: Unveiling the Value of 'r'

We've arrived at the solutions of the quadratic equation in the form x = (11 ± √101) / 2. The problem states that the solutions can also be expressed as (11 ± √r) / 2. Now, the crucial step is to compare these two expressions and determine the value of r. A direct comparison reveals a clear correspondence between the two forms. The numerators both have the term 11 ± √..., and the denominators are both 2. This means that the expressions under the square root signs must be equal. Therefore, we can directly equate √101 with √r. To find the value of r, we simply need to square both sides of this equation. Squaring √101 gives us 101, and squaring √r gives us r. Thus, we conclude that r = 101. This simple yet powerful comparison highlights the elegance of mathematical expressions and how careful observation can lead to straightforward solutions. The value of r is the number under the square root sign in the solution, and by directly comparing the derived solution with the given form, we have successfully identified it. This final step completes our journey of solving the quadratic equation and finding the value of 'r'. The process has involved several key steps: identifying the coefficients, applying the quadratic formula, simplifying the expression, and finally, comparing the result with the given form. Each step has required careful attention to detail and a solid understanding of algebraic principles. The result, r = 101, is a concrete numerical answer that represents a specific aspect of the solutions to the quadratic equation. It underscores the connection between the coefficients of the equation and the form of its roots. In conclusion, by systematically applying the quadratic formula and carefully comparing the results, we have successfully determined the value of r to be 101. This exercise not only strengthens our understanding of quadratic equations but also demonstrates the power of algebraic manipulation and the importance of paying close attention to detail in mathematical problem-solving. The journey from the initial equation to the final answer has been a testament to the beauty and elegance of mathematical reasoning.

H2: Conclusion: The Power of Quadratic Equations

In this exploration, we successfully navigated the realm of quadratic equations to determine the value of 'r' in the solution form (11 ± √r) / 2 for the equation x² - 11x + 5 = 0. This journey highlighted the power and versatility of the quadratic formula, a fundamental tool in algebra. We began by understanding the quadratic formula and its components, particularly the discriminant, which plays a crucial role in determining the nature of the roots. Then, we meticulously identified the coefficients a, b, and c from the given equation, ensuring accuracy in our substitutions. The next step involved applying the quadratic formula, substituting the identified coefficients, and simplifying the resulting expression. This process showcased the importance of careful algebraic manipulation and attention to detail. We arrived at the solutions in the form x = (11 ± √101) / 2. Finally, we compared this result with the given solution form (11 ± √r) / 2, leading us to the straightforward conclusion that r = 101. This final comparison emphasized the elegance of mathematical expressions and the power of direct observation in problem-solving. This exercise demonstrates how abstract algebraic concepts can translate into concrete numerical values. The quadratic formula is not just a formula to memorize; it's a powerful tool that allows us to solve a wide range of quadratic equations. Understanding the formula and its components is essential for success in algebra and beyond. The value of 'r', in this case, provides a specific piece of information about the solutions to the equation. It represents the number under the square root in the simplified solution form. The entire process, from identifying coefficients to making the final comparison, underscores the importance of a systematic approach to problem-solving. Each step builds upon the previous one, and accuracy at each stage is crucial for arriving at the correct answer. In conclusion, we have successfully determined the value of r to be 101 by applying the quadratic formula and carefully comparing the results. This exploration has not only solved a specific problem but also reinforced our understanding of quadratic equations and the power of algebraic reasoning. The journey through this mathematical problem has been a testament to the beauty and elegance of mathematics.

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