Solving Quadratic Equations A Step By Step Guide To X² - 10x = -34

In the realm of mathematics, quadratic equations hold a significant position, serving as fundamental tools for modeling and solving a myriad of real-world problems. These equations, characterized by their second-degree polynomial form, often present us with intriguing challenges, particularly when it comes to finding their solutions, also known as roots. In this comprehensive exploration, we embark on a journey to unravel the solutions of the quadratic equation x² - 10x = -34, delving into the intricacies of the quadratic formula and its application in determining the nature and values of the roots. Understanding the solutions of this equation not only enhances our mathematical proficiency but also equips us with the ability to tackle various problems across diverse fields, including physics, engineering, and economics. Solving quadratic equations might seem daunting at first, but with a systematic approach and a clear understanding of the underlying principles, you can confidently navigate through these mathematical challenges. So, let's dive into the world of quadratic equations and uncover the solutions to the equation x² - 10x = -34.

Before we can effectively apply the quadratic formula, it's crucial to manipulate the given equation into the standard quadratic form, which is ax² + bx + c = 0. This form allows us to readily identify the coefficients a, b, and c, which are essential components of the quadratic formula. Our starting equation is x² - 10x = -34. To transform it into the standard form, we need to move the constant term, -34, to the left side of the equation. This is achieved by adding 34 to both sides, resulting in the equation x² - 10x + 34 = 0. Now, we have successfully expressed the equation in the standard quadratic form, where a = 1, b = -10, and c = 34. With the equation in this form, we are well-prepared to employ the quadratic formula and embark on the quest to find the solutions. This transformation is a critical first step in solving any quadratic equation, as it sets the stage for the application of various solution methods, including the quadratic formula, factoring, and completing the square. By mastering this initial step, you'll be well-equipped to tackle a wide range of quadratic equation problems. The importance of this step cannot be overstated, as it provides a clear and organized foundation for the subsequent steps in the solution process.

With the equation now in its standard form, we can confidently unleash the power of the quadratic formula, a mathematical tool that provides a direct pathway to the solutions of any quadratic equation. The quadratic formula is expressed as x = (-b ± √(b² - 4ac)) / 2a. This formula might seem intimidating at first glance, but its structure is quite logical and easy to apply once you understand its components. The symbols a, b, and c represent the coefficients of the quadratic equation in its standard form, which we have already identified as a = 1, b = -10, and c = 34 for our equation. The symbol ± indicates that there are two possible solutions, one obtained by adding the square root term and the other by subtracting it. The square root term, √(b² - 4ac), is particularly important as it determines the nature of the solutions. If the value inside the square root (b² - 4ac), known as the discriminant, is positive, there are two distinct real solutions. If it's zero, there is one real solution (a repeated root). And if it's negative, there are two complex solutions. Now, let's plug the values of a, b, and c into the quadratic formula and see what solutions it unveils for our equation. This step is the heart of solving quadratic equations, and mastering the application of the quadratic formula is a crucial skill for any aspiring mathematician.

Before we fully plunge into the quadratic formula, let's take a moment to calculate the discriminant, b² - 4ac. This seemingly simple expression holds the key to understanding the nature of the solutions we are about to find. As mentioned earlier, the discriminant tells us whether the solutions are real and distinct, real and repeated, or complex. For our equation, x² - 10x + 34 = 0, we have a = 1, b = -10, and c = 34. Plugging these values into the discriminant formula, we get (-10)² - 4 * 1 * 34 = 100 - 136 = -36. The discriminant, -36, is a negative number. This immediately tells us that the solutions to our quadratic equation are complex numbers. Complex numbers involve the imaginary unit, denoted by i, which is defined as the square root of -1. When the discriminant is negative, the square root in the quadratic formula will involve the square root of a negative number, leading to complex solutions. Understanding the discriminant is a powerful tool in solving quadratic equations. It allows us to anticipate the type of solutions we will encounter, saving us time and effort in the long run. In our case, knowing that the solutions are complex prepares us to work with imaginary numbers and express the solutions in the form a + bi, where a and b are real numbers and i is the imaginary unit. This preliminary calculation provides valuable insights and guides our subsequent steps in solving the equation.

Now, armed with the knowledge that our solutions are complex, we proceed to substitute the values of a, b, and c into the quadratic formula: x = (-(-10) ± √((-10)² - 4 * 1 * 34)) / (2 * 1). This might look like a daunting expression, but we will simplify it step by step. First, we simplify the terms inside the square root. We already calculated the discriminant as -36, so we have √(-36). Since the square root of a negative number involves the imaginary unit i, we can rewrite this as √(36 * -1) = √(36) * √(-1) = 6i. Next, we simplify the other terms in the formula. -(-10) becomes 10, and 2 * 1 becomes 2. So, our equation now looks like this: x = (10 ± 6i) / 2. The final step is to divide both terms in the numerator by 2. This gives us x = 5 ± 3i. These are the two complex solutions to our quadratic equation. They are complex conjugates of each other, meaning they have the same real part (5) but opposite imaginary parts (3i and -3i). This is a common characteristic of complex solutions to quadratic equations with real coefficients. By carefully substituting the values and simplifying the expression, we have successfully navigated the complex realm and found the solutions to our equation. This process demonstrates the power and versatility of the quadratic formula in solving a wide range of quadratic equations, including those with complex solutions.

After meticulous application of the quadratic formula and careful simplification, we have arrived at the solutions to the quadratic equation x² - 10x = -34. The solutions are x = 5 + 3i and x = 5 - 3i. These are complex numbers, each consisting of a real part (5) and an imaginary part (3i and -3i, respectively). The presence of the imaginary unit i indicates that these solutions do not lie on the real number line, but rather in the complex plane. Complex solutions arise when the discriminant of the quadratic equation is negative, as we found in our calculation. The solutions are complex conjugates of each other, a common occurrence in quadratic equations with real coefficients. This means they have the same real part but opposite imaginary parts. The solutions x = 5 + 3i and x = 5 - 3i represent the values of x that satisfy the original equation. If we were to substitute either of these values back into the equation x² - 10x = -34, we would find that the equation holds true. Finding the solutions to a quadratic equation is a fundamental skill in algebra, and understanding the nature of the solutions, whether they are real or complex, is crucial for many applications in mathematics and other fields. In this case, we have successfully found the complex solutions to the equation, demonstrating our mastery of the quadratic formula and our ability to work with complex numbers.

While we have confidently arrived at the solutions x = 5 + 3i and x = 5 - 3i, it's always a good practice to verify our results. This step not only ensures the accuracy of our calculations but also deepens our understanding of how the solutions satisfy the original equation. To verify the solutions, we substitute each value back into the original equation, x² - 10x = -34, and check if the equation holds true. Let's start with x = 5 + 3i. Substituting this value into the equation, we get (5 + 3i)² - 10(5 + 3i) = -34. Expanding the terms, we have (25 + 30i - 9) - (50 + 30i) = -34. Simplifying further, we get 16 + 30i - 50 - 30i = -34, which reduces to -34 = -34. This confirms that x = 5 + 3i is indeed a solution. Now, let's verify x = 5 - 3i. Substituting this value into the equation, we get (5 - 3i)² - 10(5 - 3i) = -34. Expanding the terms, we have (25 - 30i - 9) - (50 - 30i) = -34. Simplifying further, we get 16 - 30i - 50 + 30i = -34, which reduces to -34 = -34. This confirms that x = 5 - 3i is also a solution. By verifying both solutions, we have strengthened our confidence in the accuracy of our results and gained a deeper understanding of how these complex numbers satisfy the original quadratic equation. This step is an important part of the problem-solving process, as it allows us to catch any potential errors and reinforce our understanding of the concepts involved.

In this comprehensive exploration, we have successfully navigated the intricacies of solving the quadratic equation x² - 10x = -34. We began by transforming the equation into its standard form, x² - 10x + 34 = 0, which allowed us to readily identify the coefficients a, b, and c. We then introduced the quadratic formula, the key to unlocking the solutions of any quadratic equation. Before fully applying the formula, we calculated the discriminant, b² - 4ac, which revealed that the solutions would be complex numbers. Armed with this knowledge, we substituted the values of a, b, and c into the quadratic formula and carefully simplified the expression, ultimately arriving at the solutions x = 5 + 3i and x = 5 - 3i. To ensure the accuracy of our results, we verified the solutions by substituting them back into the original equation, confirming that they indeed satisfy the equation. This journey has not only provided us with the solutions to this specific quadratic equation but has also reinforced our understanding of the quadratic formula, the discriminant, and the nature of complex numbers. Mastering the art of solving quadratic equations is a fundamental skill in mathematics, with applications spanning various fields. By understanding the concepts and techniques discussed in this exploration, you are well-equipped to tackle a wide range of quadratic equation problems and appreciate their significance in the world around us. The ability to solve quadratic equations opens doors to a deeper understanding of mathematical principles and their applications in real-world scenarios.