Solving The Quadratic Equation X² - 10x = -34
In the realm of mathematics, quadratic equations hold a significant place. These equations, characterized by the presence of a squared variable, often present a fascinating challenge to solve. The general form of a quadratic equation is expressed as ax² + bx + c = 0, where a, b, and c are constants, and 'x' represents the variable we aim to find. Solving a quadratic equation means determining the values of 'x' that satisfy the equation, also known as the roots or solutions of the equation. These solutions can be real numbers, representing points where the quadratic function intersects the x-axis, or complex numbers, indicating the absence of real roots. Several methods exist for tackling quadratic equations, each with its own strengths and suitability depending on the specific equation at hand. Factoring, a straightforward approach, involves breaking down the quadratic expression into the product of two linear factors. This method proves particularly effective when the roots are rational numbers. Completing the square, another powerful technique, transforms the equation into a perfect square trinomial, making it easier to isolate the variable 'x'. However, the most versatile and widely applicable method is the quadratic formula, a robust tool that provides solutions for any quadratic equation, regardless of the nature of its roots. This formula, derived from the process of completing the square, offers a direct pathway to the solutions, ensuring that no quadratic equation remains unsolvable. Understanding quadratic equations and mastering their solution techniques are fundamental skills in algebra and calculus, paving the way for tackling more complex mathematical concepts and real-world applications.
Exploring the Equation x² - 10x = -34
Let's dive into the specific quadratic equation at hand: x² - 10x = -34. This equation presents a classic example of a quadratic expression, where we seek to find the values of 'x' that make the equation true. Before we can unleash our arsenal of solution methods, we need to massage the equation into its standard form. The standard form, as mentioned earlier, is ax² + bx + c = 0. This form provides a clear structure for identifying the coefficients a, b, and c, which are crucial for applying methods like the quadratic formula. To transform our equation, we simply add 34 to both sides, resulting in x² - 10x + 34 = 0. Now, we have a quadratic equation neatly presented in its standard form, ready for analysis. We can now identify the coefficients: a = 1, b = -10, and c = 34. These values will be instrumental in our quest to find the solutions. Examining the equation closely, we might attempt to solve it by factoring. However, factoring often proves challenging when the roots are not rational numbers. In such cases, alternative methods like completing the square or the quadratic formula offer more reliable pathways to the solutions. The quadratic formula, in particular, stands out as a universally applicable tool, capable of handling any quadratic equation, regardless of the nature of its roots. As we proceed, we will explore the quadratic formula in detail, demonstrating its power and versatility in solving this equation and others like it. The journey to finding the solutions involves careful application of mathematical principles and a keen eye for detail.
Method 1: Applying the Quadratic Formula
The quadratic formula stands as a cornerstone in solving quadratic equations. It offers a direct and reliable method for finding the roots, regardless of whether they are real or complex. The formula itself is a compact expression derived from the process of completing the square, encapsulating the essence of quadratic equation solutions. It states that for a quadratic equation in the standard form ax² + bx + c = 0, the solutions for 'x' are given by: x = (-b ± √(b² - 4ac)) / 2a. This formula might seem intimidating at first glance, but its power lies in its ability to handle any quadratic equation with ease. The expression within the square root, b² - 4ac, is known as the discriminant. The discriminant plays a crucial role in determining the nature of the roots. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root (a repeated root). And if it is negative, the equation has two complex roots. Complex roots arise when the square root of a negative number is encountered, leading to the involvement of imaginary units. To apply the quadratic formula to our equation, x² - 10x + 34 = 0, we recall the coefficients we identified earlier: a = 1, b = -10, and c = 34. Plugging these values into the formula, we get: x = (10 ± √((-10)² - 4 * 1 * 34)) / (2 * 1). This is where the arithmetic takes center stage. We carefully evaluate the expression under the square root and simplify the entire expression to arrive at the solutions. The beauty of the quadratic formula lies in its systematic approach, guiding us step-by-step towards the solutions, regardless of the complexity of the equation.
Step-by-Step Solution using the Quadratic Formula
Let's embark on a step-by-step journey to solve the equation x² - 10x + 34 = 0 using the quadratic formula. As we established, the formula is x = (-b ± √(b² - 4ac)) / 2a. Our first step is to substitute the values of a, b, and c into the formula. We have a = 1, b = -10, and c = 34. Substituting these values, we get: x = (10 ± √((-10)² - 4 * 1 * 34)) / (2 * 1). Now, we need to simplify the expression inside the square root. (-10)² is 100, and 4 * 1 * 34 is 136. So, we have: x = (10 ± √(100 - 136)) / 2. Next, we simplify the expression under the square root: 100 - 136 = -36. This gives us: x = (10 ± √(-36)) / 2. Here, we encounter the square root of a negative number, indicating that the solutions will be complex numbers. The square root of -36 is 6i, where 'i' is the imaginary unit (√-1). Thus, we have: x = (10 ± 6i) / 2. Finally, we simplify the expression by dividing both the real and imaginary parts by 2: x = 5 ± 3i. This reveals the two solutions to the quadratic equation: x = 5 + 3i and x = 5 - 3i. These are complex conjugate roots, a common occurrence when the discriminant is negative. The step-by-step application of the quadratic formula has led us to the solutions, showcasing its methodical approach and its ability to handle complex roots with precision. The journey through the formula highlights the importance of careful arithmetic and a clear understanding of complex numbers.
Identifying the Correct Answer
Having meticulously applied the quadratic formula, we have arrived at the solutions x = 5 + 3i and x = 5 - 3i. Now, let's compare these solutions with the options provided in the question. The options are:
A. x = -8, -2 B. x = 5 ± 3i C. x = -5 ± 3i D. x = -5 ± √59
By direct comparison, we can see that our solutions, x = 5 + 3i and x = 5 - 3i, perfectly match option B. Therefore, option B is the correct answer. The other options present different values, which do not satisfy the original quadratic equation. Option A suggests two real roots, which is not the case for our equation, as we found complex roots. Options C and D also propose different sets of roots, none of which align with our calculated solutions. The process of identifying the correct answer underscores the importance of careful calculation and comparison. A single error in the application of the quadratic formula could lead to an incorrect result. Similarly, a hasty glance at the options without proper verification could lead to selecting the wrong answer. The meticulous approach we have taken, from transforming the equation to applying the formula and simplifying the results, ensures that we arrive at the correct solution with confidence. The final step of comparing our solutions with the options serves as a crucial check, reinforcing the accuracy of our work and solidifying our understanding of the problem-solving process.
Conclusion: The Solutions are Complex
In conclusion, the solutions to the quadratic equation x² - 10x = -34 are complex numbers, specifically x = 5 + 3i and x = 5 - 3i. We arrived at these solutions by employing the powerful quadratic formula, a versatile tool that handles both real and complex roots with ease. The journey involved transforming the equation into its standard form, identifying the coefficients, substituting them into the formula, and carefully simplifying the resulting expression. The discriminant, b² - 4ac, played a crucial role in revealing the nature of the roots. In this case, the negative discriminant indicated the presence of complex conjugate roots. The step-by-step application of the quadratic formula showcased its methodical approach, guiding us through the calculations with precision. The final step of comparing our solutions with the options provided confirmed that option B, x = 5 ± 3i, is the correct answer. This exercise highlights the importance of understanding quadratic equations, mastering solution techniques, and paying meticulous attention to detail. Quadratic equations are fundamental in mathematics, appearing in various contexts and applications. The ability to solve them efficiently and accurately is a valuable skill. The quadratic formula, in particular, stands as a cornerstone in the toolbox of any mathematician, providing a reliable pathway to the solutions, regardless of the complexity of the equation. The exploration of this equation has not only provided us with the solutions but also reinforced our understanding of quadratic equations and the power of the quadratic formula.
