Solutions To 4x² - 40x + 109 = 0 A Comprehensive Guide
This article delves into the process of finding the solutions for the given quadratic equation, 4x² - 40x + 109 = 0. Quadratic equations, a fundamental concept in mathematics, often appear in various real-world applications, making it crucial to understand their solutions. We will explore the standard methods for solving quadratic equations, with a particular focus on the quadratic formula, to accurately determine the values of x that satisfy the equation. Through a step-by-step approach, this discussion aims to provide a clear and concise understanding of how to tackle such problems, ensuring readers can confidently solve similar quadratic equations in the future. The importance of mastering these techniques extends beyond the classroom, as quadratic equations frequently arise in physics, engineering, economics, and computer science, highlighting their broad applicability and the need for a solid grasp of their solutions. Understanding the nature of the solutions, whether they are real or complex, is equally important and will be thoroughly addressed in our analysis. By the end of this exploration, you will have a comprehensive understanding of how to apply the quadratic formula and interpret the results, solidifying your foundation in algebra and mathematical problem-solving.
Understanding Quadratic Equations
Before diving into the specifics of solving the equation 4x² - 40x + 109 = 0, it's essential to grasp the fundamental nature of quadratic equations. A quadratic equation is a polynomial equation of the second degree, meaning it includes at least one term that is squared. The standard form of a quadratic equation is expressed as ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation, also known as roots or zeros, are the values of x that make the equation true. These solutions represent the points where the parabola defined by the quadratic equation intersects the x-axis when graphed. Quadratic equations can have two distinct real solutions, one repeated real solution, or two complex solutions, depending on the discriminant (b² - 4ac). Understanding the discriminant is crucial because it determines the nature of the roots. A positive discriminant indicates two distinct real roots, a zero discriminant indicates one repeated real root, and a negative discriminant indicates two complex roots. The coefficients a, b, and c play significant roles in determining the shape and position of the parabola, as well as the solutions to the equation. Recognizing these foundational aspects of quadratic equations sets the stage for effectively applying methods like the quadratic formula to find the solutions. This understanding is not only crucial for academic success but also for practical applications where quadratic equations model various phenomena, from projectile motion in physics to optimization problems in economics.
Method 1: Applying the Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form ax² + bx + c = 0. This formula provides a direct method for finding the solutions, regardless of whether the roots are real or complex. The quadratic formula is given by: x = (-b ± √(b² - 4ac)) / (2a). In this formula, a, b, and c are the coefficients from the quadratic equation. The term inside the square root, b² - 4ac, is known as the discriminant, which, as discussed earlier, determines the nature of the solutions. When applying the quadratic formula, the first step involves identifying the values of a, b, and c from the given equation. Once these values are determined, they are substituted into the formula. The next step involves simplifying the expression under the square root (the discriminant) and then proceeding with the remaining calculations. The ± sign in the formula indicates that there are typically two solutions: one obtained by adding the square root term and the other by subtracting it. These two solutions may be distinct, equal, or complex conjugates, depending on the value of the discriminant. The quadratic formula is particularly useful when factoring the quadratic equation is difficult or not possible. Its versatility and reliability make it a fundamental technique in algebra. Furthermore, understanding the derivation and application of the quadratic formula provides a deeper insight into the mathematical structure of quadratic equations and their solutions. Mastering this method is essential for students and professionals alike, as it is widely used in various fields requiring mathematical modeling and problem-solving.
Solving 4x² - 40x + 109 = 0 Using the Quadratic Formula
Now, let's apply the quadratic formula to solve the specific equation 4x² - 40x + 109 = 0. First, we need to identify the coefficients a, b, and c. In this case, a = 4, b = -40, and c = 109. Plugging these values into the quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), we get: x = (-(-40) ± √((-40)² - 4 * 4 * 109)) / (2 * 4). Next, we simplify the expression step by step. The numerator becomes 40 ± √(1600 - 1744), and the denominator is 8. Further simplification inside the square root gives us √(1600 - 1744) = √(-144). Since the value inside the square root is negative, we know the solutions will be complex. The square root of -144 can be expressed as 12i, where i is the imaginary unit (√-1). So, the equation becomes x = (40 ± 12i) / 8. Now, we can divide both terms in the numerator by the denominator, 8, to get x = 5 ± (3/2)i. This result provides the two complex solutions for the quadratic equation. Understanding how to handle complex numbers is crucial when solving quadratic equations with negative discriminants. The ability to apply the quadratic formula correctly and simplify the resulting expressions ensures accurate solutions, even when dealing with complex roots. This step-by-step approach highlights the importance of methodical calculation and attention to detail in solving quadratic equations.
Analyzing the Solutions: Complex Roots
In the previous section, we found that the solutions to the quadratic equation 4x² - 40x + 109 = 0 are complex numbers, specifically x = 5 ± (3/2)i. This outcome is a direct consequence of the negative discriminant we encountered during the application of the quadratic formula. A negative discriminant, b² - 4ac < 0, indicates that the roots of the quadratic equation are not real numbers but rather complex conjugates. Complex numbers have the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1). In our solutions, the real part is 5, and the imaginary part is ±(3/2). The presence of complex roots has significant implications in various fields. For instance, in electrical engineering, complex numbers are used to describe alternating current circuits, where the imaginary part represents the phase difference between voltage and current. In quantum mechanics, complex numbers are fundamental in describing wave functions. Geometrically, complex roots mean that the parabola represented by the quadratic equation does not intersect the x-axis. This is because the solutions represent the x-intercepts of the parabola, and complex solutions indicate that the parabola does not cross the x-axis. Understanding the nature of complex roots and their significance is crucial for a comprehensive understanding of quadratic equations. It extends the applicability of quadratic equations beyond scenarios with real solutions, allowing us to model and analyze a broader range of phenomena. Furthermore, the conjugate nature of complex roots (i.e., a + bi and a - bi) is a consistent property that arises from the quadratic formula, highlighting the symmetry inherent in quadratic equations.
Choosing the Correct Answer
After solving the quadratic equation 4x² - 40x + 109 = 0 using the quadratic formula, we arrived at the solutions x = 5 ± (3/2)i. Now, we need to identify which of the given options matches our result. The options provided are:
A. x = -5 ± (3/2)i B. x = 5 ± (3/2)i C. x = -5 ± 12i D. x = 5 ± 12i
By comparing our calculated solutions with the options, it's clear that option B, x = 5 ± (3/2)i, exactly matches our result. The other options either have the wrong sign for the real part (-5 instead of 5) or an incorrect imaginary part (12i instead of (3/2)i). This step underscores the importance of careful calculation and attention to detail throughout the problem-solving process. A minor error in applying the quadratic formula or simplifying the expressions can lead to an incorrect answer. Therefore, it's crucial to double-check each step to ensure accuracy. In the context of multiple-choice questions, matching the calculated solution with the available options is a final check that confirms the correctness of the approach and the result. This process reinforces the understanding of the problem and the solution method. It also highlights the value of practicing similar problems to build confidence and proficiency in solving quadratic equations.
Conclusion
In conclusion, we have successfully solved the quadratic equation 4x² - 40x + 109 = 0 using the quadratic formula. By identifying the coefficients a, b, and c, applying the formula x = (-b ± √(b² - 4ac)) / (2a), and simplifying the resulting expression, we found the solutions to be x = 5 ± (3/2)i. These solutions are complex numbers, which was indicated by the negative discriminant (b² - 4ac). Understanding the nature of the discriminant and its implications for the types of solutions (real or complex) is a crucial aspect of solving quadratic equations. The quadratic formula provides a reliable method for finding solutions, regardless of whether the roots are real or complex. This technique is not only essential for academic mathematics but also for practical applications in various fields, including physics, engineering, and computer science. The step-by-step approach outlined in this discussion emphasizes the importance of methodical calculation and attention to detail. From identifying coefficients to simplifying expressions involving complex numbers, each step contributes to the accuracy of the final result. Moreover, the ability to interpret the solutions and understand their significance, such as recognizing complex roots and their implications, enhances the overall problem-solving skills. By mastering these techniques, individuals can confidently tackle a wide range of quadratic equation problems and apply them effectively in real-world scenarios.
