Solving R² - 5r + 25 = 0 A Step-by-Step Guide

In this comprehensive guide, we will delve into the process of solving the quadratic equation r² - 5r + 25 = 0. Quadratic equations, which take the general form ax² + bx + c = 0, are fundamental in mathematics and appear in various fields, including physics, engineering, and computer science. Our focus here will be on finding the solutions, also known as roots, of the given equation. These roots represent the values of 'r' that satisfy the equation, making the expression equal to zero. We will explore the standard methods for solving quadratic equations, paying close attention to the complexities that arise when dealing with non-real, or complex, solutions. This step-by-step approach ensures a clear understanding of the underlying concepts and techniques involved in solving such equations, making it accessible for learners of all levels. Understanding quadratic equations is crucial for further studies in algebra and related fields.

There are several methods to solve quadratic equations, but we will focus on the quadratic formula, which is the most versatile method and can be applied to any quadratic equation, regardless of whether its roots are real or complex. Other methods, such as factoring and completing the square, are also useful but may not be as universally applicable. Factoring, for example, is efficient when the quadratic expression can be easily factored into two binomials. However, many quadratic equations do not factor easily, making the quadratic formula the preferred method. Completing the square is another method that can be used to solve quadratic equations, and it also forms the basis for deriving the quadratic formula itself. This method involves manipulating the equation to create a perfect square trinomial, which can then be easily solved. However, completing the square can be more cumbersome than the quadratic formula, especially when the coefficients are not simple integers. Therefore, for the equation r² - 5r + 25 = 0, we will use the quadratic formula to ensure accuracy and efficiency. This method not only provides the solutions but also gives insight into the nature of the roots, whether they are real, complex, rational, or irrational. By mastering the quadratic formula, you gain a powerful tool for solving a wide range of mathematical problems.

The quadratic formula is given by: r = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0. In our case, the equation is r² - 5r + 25 = 0, so we can identify a = 1, b = -5, and c = 25. The heart of the quadratic formula lies in the discriminant, which is the expression b² - 4ac. The discriminant provides valuable information about the nature of the roots of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has exactly one real root (a repeated root). And if the discriminant is negative, the equation has two complex conjugate roots. Substituting the values of a, b, and c into the discriminant, we get (-5)² - 4(1)(25) = 25 - 100 = -75. Since the discriminant is negative, we know that the equation has two complex conjugate roots. Now, we proceed with the full quadratic formula: r = (-(-5) ± √((-5)² - 4(1)(25))) / (2(1)). This simplifies to r = (5 ± √(-75)) / 2. We can further simplify the square root of -75 by expressing it in terms of the imaginary unit 'i', where i = √(-1). Thus, √(-75) = √(75 * -1) = √(75) * √(-1) = 5√3 * i. Substituting this back into the formula, we get r = (5 ± 5√3i) / 2. This gives us two complex solutions for r: r = (5 + 5√3i) / 2 and r = (5 - 5√3i) / 2. These are the fully simplified, non-real solutions to the given quadratic equation. Understanding the discriminant is key to predicting the nature of the roots, and the quadratic formula is the ultimate tool for finding these roots.

To fully simplify the solutions, we express them in the standard form of complex numbers, which is a + bi, where 'a' is the real part and 'b' is the imaginary part. From the previous step, we have r = (5 ± 5√3i) / 2. We can separate this into two distinct solutions: r₁ = (5 + 5√3i) / 2 and r₂ = (5 - 5√3i) / 2. To express r₁ in the form a + bi, we divide both the real and imaginary parts by 2: r₁ = 5/2 + (5√3/2)i. Similarly, for r₂, we divide both the real and imaginary parts by 2: r₂ = 5/2 - (5√3/2)i. These are the simplified complex solutions for the quadratic equation r² - 5r + 25 = 0. Notice that the solutions are complex conjugates of each other, which is a characteristic of quadratic equations with a negative discriminant. Complex conjugate pairs always have the same real part but opposite imaginary parts. In this case, the real part is 5/2, and the imaginary parts are (5√3/2)i and -(5√3/2)i. The solutions are now in their simplest form, making it easy to identify the real and imaginary components. Simplifying complex numbers is an essential skill in advanced mathematics and engineering. These simplified solutions are ready for use in any further calculations or applications where the roots of the quadratic equation are needed.

In conclusion, by applying the quadratic formula to the equation r² - 5r + 25 = 0, we have found the solutions to be complex numbers. These solutions, when fully simplified, are r = 5/2 + (5√3/2)i and r = 5/2 - (5√3/2)i. These complex solutions indicate that the parabola represented by the quadratic equation does not intersect the real number line, which corresponds to the x-axis when graphed. The discriminant, being negative, confirmed this before we even completed the full calculation. The importance of understanding complex solutions cannot be overstated, as they appear frequently in more advanced mathematical contexts and in various scientific and engineering applications. From electrical engineering to quantum mechanics, complex numbers provide a necessary tool for modeling and understanding phenomena that cannot be described by real numbers alone. This detailed walkthrough provides a clear understanding of how to solve quadratic equations, including those with complex roots, and demonstrates the power and versatility of the quadratic formula. The process involves identifying the coefficients, applying the formula, simplifying the results, and expressing the solutions in their simplest form. Mastering this process is a significant step in your mathematical journey.

In summary, we have successfully solved the quadratic equation r² - 5r + 25 = 0 using the quadratic formula. We identified the coefficients, calculated the discriminant, and found the complex solutions. The solutions are r = 5/2 + (5√3/2)i and r = 5/2 - (5√3/2)i. This exercise demonstrates the effectiveness of the quadratic formula in finding both real and complex roots of quadratic equations. The negative discriminant alerted us to the presence of complex solutions, which we then expressed in their simplified a + bi form. This comprehensive guide equips you with the knowledge and skills to tackle similar quadratic equations with confidence. Remember, practice is key to mastering these techniques. Try solving other quadratic equations with both real and complex roots to solidify your understanding. The quadratic formula is your reliable companion in the world of algebra, and understanding its nuances will serve you well in your future mathematical endeavors.