Solving (x - R)^2 = (2c - R)(x^2 - R) Finding The Value Of C

Introduction: Delving into the Depths of Quadratic Equations

At the heart of algebra lies the quadratic equation, a fundamental concept with far-reaching applications in various fields, from physics and engineering to economics and computer science. This article embarks on a journey to dissect a specific quadratic equation, (x - r)^2 = (2c - r)(x^2 - r), where c and r are constants with the condition r < 2c. Our mission is to unravel the complexities of this equation, ultimately aiming to determine the elusive value of c, given that the product of the solutions to the equation is 5r - 12. We'll begin by expanding and rearranging the equation to bring it into the standard quadratic form, ax^2 + bx + c = 0. This transformation will allow us to apply the powerful tools and techniques associated with quadratic equations, such as the quadratic formula and Vieta's formulas. Along the way, we'll explore the significance of the given condition, r < 2c, and how it influences the nature of the solutions. Furthermore, we'll leverage the information about the product of the solutions to establish a crucial relationship between c and r, paving the way for the final determination of c. This exploration will not only enhance our understanding of quadratic equations but also showcase the elegance and interconnectedness of mathematical concepts. As we navigate through the intricacies of this problem, we'll emphasize clarity and precision, ensuring that each step is thoroughly explained and justified. Our goal is to provide a comprehensive and insightful analysis, empowering readers to confidently tackle similar challenges in the realm of algebra and beyond. So, let's embark on this mathematical adventure, where we'll uncover the hidden value of c and appreciate the beauty of quadratic equations.

Transforming the Equation: From Complexity to Clarity

To truly understand and solve the equation (x - r)^2 = (2c - r)(x^2 - r), we must first embark on a journey of transformation. The initial form of the equation, while presenting the core relationship, obscures the underlying quadratic structure. Our primary goal in this section is to manipulate the equation, step by meticulous step, until it conforms to the familiar standard form of a quadratic equation: ax^2 + bx + c = 0. This transformation is not merely an algebraic exercise; it's a crucial process that unlocks the equation's secrets and allows us to apply the powerful tools associated with quadratic equations. The journey begins with expanding both sides of the equation. On the left-hand side, we encounter the square of a binomial, (x - r)^2, which, when expanded, yields x^2 - 2rx + r^2. This expansion is a fundamental algebraic identity that we'll utilize extensively. On the right-hand side, we face the product of a binomial, (2c - r), and another expression, (x^2 - r). To expand this, we apply the distributive property, meticulously multiplying each term in the first binomial by each term in the second. This process results in (2c - r)(x^2 - r) = 2cx^2 - 2cr - rx^2 + r^2. Now, our equation takes the form x^2 - 2rx + r^2 = 2cx^2 - 2cr - rx^2 + r^2. While this equation is more expanded, it's still not in the standard quadratic form. The next step involves gathering like terms and strategically rearranging them. Our aim is to group the terms with x^2, the terms with x, and the constant terms. This process is akin to organizing a cluttered room, bringing order and clarity to the equation. Subtracting x^2 from both sides, we get -2rx = 2cx^2 - x^2 - rx^2 - 2cr. Then, adding 2rx to both sides gives 0 = 2cx^2 - x^2 - rx^2 + 2rx - 2cr. Further rearrangement leads us to the form (2c - 1 - r)x^2 + 2rx - 2cr = 0. Finally, we have arrived at the standard quadratic form, where the coefficient of x^2 is (2c - 1 - r), the coefficient of x is 2r, and the constant term is -2cr. This transformation is a significant milestone in our journey. We have successfully converted the initial equation into a recognizable quadratic form, paving the way for the application of powerful techniques to solve for the unknown variables. The coefficients we've identified, (2c - 1 - r), 2r, and -2cr, will play crucial roles in the subsequent steps of our analysis.

Applying Vieta's Formulas: Unveiling the Relationship Between Roots and Coefficients

Now that we've successfully transformed our equation into the standard quadratic form, ax^2 + bx + c = 0, where a = (2c - 1 - r), b = 2r, and c = -2cr, we can leverage the power of Vieta's formulas. These formulas provide a direct link between the coefficients of a polynomial equation and the sums and products of its roots. In the context of our quadratic equation, Vieta's formulas offer a particularly elegant and efficient way to extract valuable information. Vieta's formulas state that for a quadratic equation ax^2 + bx + c = 0, the sum of the roots (let's call them x1 and x2) is given by -b/ a, and the product of the roots is given by c/ a. These formulas are not mere mathematical curiosities; they are powerful tools that can simplify complex problems. In our case, we are particularly interested in the product of the roots, as the problem statement provides us with the crucial piece of information that the product of the solutions is 5r - 12. Applying Vieta's formulas to our equation, we find that the product of the roots, x1 * x2, is equal to (-2cr*) / (2c - 1 - r). We are given that this product is also equal to 5r - 12. Therefore, we can establish the following equation: (-2cr) / (2c - 1 - r) = 5r - 12. This equation is a pivotal point in our analysis. It connects the constants c and r through the product of the roots, a piece of information provided externally. Solving this equation will bring us closer to determining the value of c, which is our ultimate goal. To solve this equation, we'll begin by multiplying both sides by the denominator (2c - 1 - r), yielding -2cr = (5r - 12)(2c - 1 - r). This step eliminates the fraction and prepares the equation for further algebraic manipulation. Next, we'll expand the right-hand side of the equation by applying the distributive property. This expansion will result in -2cr = 10cr - 5r - 5r^2 - 24c + 12 + 12r. This expanded form reveals a more complex relationship between c and r, but it also provides us with the opportunity to rearrange the terms and potentially isolate c. We'll continue our algebraic journey in the next section, where we'll simplify this equation and solve for c. The application of Vieta's formulas has provided us with a crucial equation, and we are now poised to extract the value of c from it.

Solving for c: The Algebraic Finale

Having established the equation -2cr = 10cr - 5r - 5r^2 - 24c + 12 + 12r, our focus now shifts to the meticulous task of solving for c. This requires a series of algebraic manipulations aimed at isolating c on one side of the equation. The process may seem intricate, but each step is a logical progression, guided by the principles of algebra. First, let's gather all the terms involving c on one side of the equation. Adding 2cr to both sides, we get 0 = 12cr - 5r - 5r^2 - 24c + 12 + 12r. Next, we rearrange the terms to group the c terms together: 24c - 12cr = -5r - 5r^2 + 12 + 12r. Now, we can factor out c from the left-hand side: c(24 - 12r) = -5r^2 + 7r + 12. This factorization is a crucial step, as it isolates c within a single term. To finally solve for c, we divide both sides of the equation by (24 - 12r), yielding c = (-5r^2 + 7r + 12) / (24 - 12r). We can simplify this expression further by factoring out a common factor of 12 from the denominator: c = (-5r^2 + 7r + 12) / (12(2 - r)). Now, let's turn our attention to the numerator, -5r^2 + 7r + 12. We can attempt to factor this quadratic expression. After some careful consideration, we find that it factors into -(5r + 8)(r - 3). Therefore, our equation for c becomes c = -(5r + 8)(r - 3) / (12(2 - r)). At this point, we have an expression for c in terms of r. However, we still need to determine the specific value of c. To do this, we need to utilize the condition given in the problem statement: r < 2c. We can substitute our expression for c into this inequality and attempt to solve for r. Substituting, we get r < 2*(-(5r + 8)(r - 3) / (12(2 - r))). This inequality is more complex than our previous equation, but it provides us with the final piece of the puzzle. Simplifying the inequality, we get r < -(5r + 8)(r - 3) / (6(2 - r)). Multiplying both sides by 6(2 - r), we need to be mindful of the sign. Since r < 2c, and we expect c to be positive, it's reasonable to assume r < 2, so (2 - r) is positive. This gives us 6r(2 - r) < -(5r + 8)(r - 3). Expanding both sides, we get 12r - 6r^2 < -5r^2 + 7r + 24. Rearranging the terms, we get 0 < r^2 - 5r + 24. This quadratic inequality doesn't factor easily, and its discriminant is negative, indicating that it's always positive. Therefore, this inequality doesn't give us a specific value for r. However, we can try substituting r = 3 into our equation for c. If we do this, the numerator becomes -(5(3) + 8)(3 - 3) = 0, which would make c = 0. But if c = 0, then the original equation becomes (x - 3)^2 = 3x^2, which simplifies to 2x^2 + 6x - 9 = 0. The product of the roots of this equation is -9/2, which should equal 5(3) - 12 = 3. This is a contradiction, so r cannot be 3. Let's try to solve -5r^2 + 7r + 12 = 0 instead. This gives us r = 3 or r = -8/5. Since r != 3, r = -8/5. Now, we plug r = -8/5 into c = (-5r^2 + 7r + 12) / (24 - 12r): c = (-5(-8/5)^2 + 7(-8/5) + 12) / (24 - 12(-8/5)) c = (-64/5 - 56/5 + 60/5) / (24 + 96/5) c = (-60/5) / (216/5) c = -60 / 216 c = -5/18 This is not correct, let's review Vieta's formula: Product of root = (-2cr) / (2c - 1 - r) = 5r - 12 -2cr = (5r - 12)(2c - 1 - r) -2cr = 10cr - 5r - 5r^2 - 24c + 12 + 12r 0 = 12cr - 5r^2 + 7r - 24c + 12 24c - 12cr = -5r^2 + 7r + 12 c(24 - 12r) = -5r^2 + 7r + 12 c = (-5r^2 + 7r + 12) / (24 - 12r) c = (-5r^2 + 7r + 12) / 12(2 - r) c = -(5r + 8)(r - 3) / 12(2 - r) If r = 3: c = 0 5r - 12 = 15 - 12 = 3 if c = 0: x^2 - 2rx + r^2 = 0 (2c - r)(x^2 - r) = -r(x^2 - r) x^2 - 2*3x + 3^2 = -3(x^2 - 3) x^2 - 6x + 9 = -3x^2 + 9 4x^2 - 6x = 0 2x(2x - 3) = 0 x = 0, x = 3/2 product = 0 != 3 So r != 3 if r = -8/5: c = (-5(-8/5)^2 + 7(-8/5) + 12) / 12(2 - (-8/5)) c = (-5(64/25) - 56/5 + 12) / 12(18/5) c = (-64/5 - 56/5 + 60/5) / (216/5) c = (-60/5) / (216/5) c = -12 / (216) c = -1/18 So c = -1/18 is the answer.

Conclusion: A Journey Through Algebra's Landscape

Our exploration of the quadratic equation (x - r)^2 = (2c - r)(x^2 - r) has been a journey through the landscape of algebra, showcasing the power of algebraic manipulation, the elegance of Vieta's formulas, and the importance of careful analysis. We began by transforming the equation into the standard quadratic form, a crucial step that unlocked the equation's underlying structure. This transformation allowed us to apply Vieta's formulas, which provided a vital link between the coefficients of the equation and the product of its roots. By leveraging the information about the product of the solutions, we established an equation connecting the constants c and r. Solving this equation required a series of algebraic steps, including factoring and rearranging terms. The process was intricate, but each step was guided by the principles of algebra, ultimately leading us to an expression for c in terms of r. The condition r < 2c provided a crucial constraint, allowing us to narrow down the possible values of c. Through careful analysis and substitution, we determined that c = -1/18 . This result is the culmination of our journey, a testament to the power of mathematical reasoning and problem-solving. The exploration of this quadratic equation has not only provided us with a specific answer but has also deepened our understanding of algebraic concepts and techniques. The skills and insights gained in this process can be applied to a wide range of mathematical problems, empowering us to tackle future challenges with confidence and competence. As we conclude this article, we reflect on the beauty and interconnectedness of mathematics. The journey from the initial equation to the final solution has been a rewarding experience, highlighting the power of algebra to reveal hidden relationships and uncover profound truths. The value of c that we have found is not just a number; it is a symbol of our perseverance, our understanding, and our ability to navigate the complexities of the mathematical world.