Solving −𝑥² − 4𝑥 + 5 = 0 Find The Solution Set

Understanding quadratic equations is fundamental in mathematics. These equations, characterized by the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0, appear in various scientific and engineering applications. Solving them involves finding the values of x that satisfy the equation, known as the roots or solutions. One such equation is the focus of our attention: −x² − 4x + 5 = 0. This comprehensive guide will explore the step-by-step process of solving this equation, ensuring a clear understanding of each stage and its underlying principles. Solving this quadratic equation, −x² − 4x + 5 = 0, requires a systematic approach. The coefficients a, b, and c play a crucial role in determining the nature and value of the roots. In this case, a = -1, b = -4, and c = 5. These coefficients are the keys to unlocking the solutions using various methods, such as factoring, completing the square, or the quadratic formula. Each method offers a unique perspective on the equation, but they all converge on the same solutions. We will delve into the quadratic formula, a powerful tool applicable to any quadratic equation, ensuring a robust method for finding the solutions. Through this exploration, we aim to not only find the solutions but also to understand why and how they arise from the equation's structure. The journey through this equation is a microcosm of the broader world of quadratic equations, providing valuable insights into mathematical problem-solving and the elegant relationships between numbers and symbols. Understanding the coefficients in the equation −x² − 4x + 5 = 0 is the first step toward finding the solutions. The coefficient a = -1 indicates that the parabola opens downward, which influences the nature of the solutions. The coefficient b = -4 provides information about the axis of symmetry and the horizontal position of the parabola's vertex. The constant term c = 5 represents the y-intercept, which is the point where the parabola intersects the y-axis. These coefficients are not just numbers; they are the building blocks that shape the quadratic equation's graph and determine its solutions. The relationship between these coefficients and the solutions is at the heart of quadratic equation theory. By understanding this relationship, we can predict the nature of the solutions and choose the appropriate method to find them. Whether we use the quadratic formula, factoring, or completing the square, the coefficients guide our steps and ensure that we arrive at the correct answers. The importance of each coefficient is further amplified when we consider real-world applications of quadratic equations. From projectile motion in physics to optimization problems in economics, the coefficients represent physical quantities or constraints that dictate the system's behavior. Therefore, a solid grasp of these coefficients is essential for both mathematical understanding and practical problem-solving.

The Quadratic Formula: A Universal Solver

The quadratic formula is a cornerstone of algebra, providing a direct method to find the solutions of any quadratic equation. It is expressed as: x = [−b ± √(b² − 4ac)] / (2a). Applying this formula to our equation, we identify a = -1, b = -4, and c = 5. Substituting these values into the formula, we get: x = [−(−4) ± √((−4)² − 4(−1)(5))] / (2(−1)). This simplifies to x = [4 ± √(16 + 20)] / (−2), which further reduces to x = [4 ± √36] / (−2). The square root of 36 is 6, so we have x = [4 ± 6] / (−2). This leads to two possible solutions. The first solution, x’, is obtained by using the plus sign: x’ = (4 + 6) / (−2) = 10 / (−2) = −5. The second solution, x”, is obtained by using the minus sign: x” = (4 − 6) / (−2) = (−2) / (−2) = 1. Thus, the solutions to the equation −x² − 4x + 5 = 0 are x’ = −5 and x” = 1. The quadratic formula is not just a mathematical tool; it is a testament to the power of algebraic manipulation. Its derivation involves completing the square on the general form of a quadratic equation, a process that transforms the equation into a form where the solutions can be directly read off. This formula is universally applicable, meaning it can solve any quadratic equation, regardless of the complexity of the coefficients. It is a reliable method that guarantees finding the solutions, even when factoring or other techniques are not straightforward. In addition to its problem-solving prowess, the quadratic formula offers insights into the nature of the solutions. The discriminant, b² − 4ac, which appears under the square root, determines the type of solutions the equation has. If the discriminant is positive, the equation has two distinct real solutions. If it is zero, the equation has exactly one real solution (a repeated root). If it is negative, the equation has two complex solutions. This connection between the discriminant and the solutions highlights the deep structure of quadratic equations and the power of the quadratic formula in revealing this structure. The beauty of the quadratic formula lies in its ability to distill a complex problem into a simple, mechanical process. By plugging in the coefficients, we can bypass the intricacies of factoring or completing the square and arrive at the solutions directly. This efficiency makes it an indispensable tool for mathematicians, scientists, and engineers alike. Whether we are solving a theoretical problem or a practical application, the quadratic formula provides a reliable and elegant way to find the solutions of quadratic equations. The versatility of the quadratic formula extends beyond just finding numerical solutions. It also plays a crucial role in understanding the behavior of quadratic functions and their graphs. The solutions, also known as the roots or zeros, represent the points where the parabola intersects the x-axis. These points, along with the vertex of the parabola, provide a complete picture of the function's graph. The quadratic formula can also be used to find the axis of symmetry, which is a vertical line that divides the parabola into two symmetrical halves. The x-coordinate of the vertex lies on this line, and its value can be found using the formula x = −b / (2a), which is derived from the quadratic formula. Moreover, the quadratic formula is instrumental in solving optimization problems, where we seek to find the maximum or minimum value of a quadratic function. The vertex of the parabola represents either the maximum or minimum point, depending on the sign of the coefficient a. By using the quadratic formula to find the roots and the vertex, we can analyze the function's behavior and determine its optimal values. In real-world applications, the quadratic formula is used to model various phenomena, from the trajectory of a projectile to the shape of a suspension bridge. Its ability to provide precise solutions and insights makes it an essential tool for engineers, physicists, and other professionals. The quadratic formula is not just a formula; it is a key to unlocking the secrets of quadratic equations and their applications. Its elegance, versatility, and power make it a cornerstone of mathematical knowledge.

Verifying the Solutions: Ensuring Accuracy

After obtaining the solutions x’ = −5 and x” = 1, it is essential to verify their accuracy. To verify the solutions, we substitute each value back into the original equation, −x² − 4x + 5 = 0. For x’ = −5, we have −(−5)² − 4(−5) + 5 = −25 + 20 + 5 = 0, which confirms that x’ = −5 is a solution. For x” = 1, we have −(1)² − 4(1) + 5 = −1 − 4 + 5 = 0, which confirms that x” = 1 is also a solution. This verification process ensures that we have correctly applied the quadratic formula and have not made any arithmetic errors. Verifying the solutions is a crucial step in any mathematical problem-solving process. It provides a safety net, ensuring that we have arrived at the correct answers. In the case of quadratic equations, verification is particularly important because the solutions often have real-world interpretations. For example, in physics, the solutions might represent the time at which a projectile hits the ground. In such cases, accuracy is paramount, and verification can prevent costly errors. The process of substituting the solutions back into the original equation not only confirms their correctness but also deepens our understanding of the equation itself. It reinforces the relationship between the solutions and the coefficients, and it highlights the fact that the solutions are the values of x that make the equation true. This understanding is essential for developing a strong foundation in algebra and for applying quadratic equations to more complex problems. Furthermore, verification can reveal subtle errors that might otherwise go unnoticed. For instance, if we had made a mistake in applying the quadratic formula, the verification process would show that the substituted values do not satisfy the equation. This immediate feedback allows us to correct the error and learn from our mistakes. In summary, verifying the solutions is not just a formality; it is an integral part of the problem-solving process. It ensures accuracy, deepens understanding, and provides valuable feedback that enhances our mathematical skills. By making verification a habit, we can approach mathematical problems with confidence and ensure that our solutions are reliable. The practice of verifying solutions extends beyond quadratic equations to all areas of mathematics. Whether we are solving linear equations, trigonometric equations, or differential equations, the principle remains the same: substitute the solutions back into the original equation to ensure that they satisfy it. This simple step can save us from significant errors and build our confidence in our problem-solving abilities. In the context of technology, verification has become even more critical. With the widespread use of calculators and computer software, it is easy to generate solutions without fully understanding the underlying mathematics. Verification allows us to check the results produced by these tools and ensure that they are correct. It also helps us to develop a critical eye for the output of technology, preventing us from blindly accepting results without understanding their validity. In educational settings, verification should be emphasized as a key component of mathematical literacy. Students should be encouraged to verify their solutions and to explain why this process is important. This not only reinforces their understanding of the mathematical concepts but also cultivates a habit of critical thinking and attention to detail. Verification is not just about finding the right answer; it is about developing a deeper understanding of the mathematics and building the skills necessary to solve problems effectively. By making verification a routine part of our problem-solving process, we can enhance our mathematical abilities and approach challenges with confidence and precision.

Conclusion: The Solutions Unveiled

In conclusion, by applying the quadratic formula to the equation −x² − 4x + 5 = 0, we have successfully determined the solutions to be x’ = −5 and x” = 1. These solutions represent the values of x that satisfy the equation, and their accuracy has been verified through substitution. This process demonstrates the power and versatility of the quadratic formula in solving quadratic equations. Solving this equation, −x² − 4x + 5 = 0, has been a journey through the core concepts of quadratic equations. We started by understanding the coefficients and their significance, then applied the quadratic formula to find the solutions, and finally verified the solutions to ensure accuracy. This step-by-step approach highlights the importance of a systematic method in mathematical problem-solving. The solutions x’ = −5 and x” = 1 are not just numbers; they are the points where the parabola represented by the equation intersects the x-axis. They provide a visual representation of the equation's behavior and its relationship to the coordinate plane. Understanding these solutions is crucial for interpreting the equation in real-world contexts, such as physics, engineering, and economics. The process of solving this equation has also reinforced the importance of attention to detail and careful execution. A small error in applying the quadratic formula or in the arithmetic calculations can lead to incorrect solutions. Verification serves as a crucial check, ensuring that we have not made any mistakes. This emphasis on accuracy is a valuable lesson that extends beyond mathematics to all areas of life. Furthermore, this exercise has demonstrated the power of algebraic manipulation in transforming a complex equation into a form where the solutions can be easily found. The quadratic formula is a testament to the ingenuity of mathematicians and their ability to distill complex problems into elegant solutions. By mastering this formula and the techniques for solving quadratic equations, we gain a powerful tool for tackling a wide range of mathematical challenges. The journey through this equation has also highlighted the interconnectedness of mathematical concepts. Quadratic equations are not isolated entities; they are part of a larger web of mathematical ideas, including linear equations, polynomials, and calculus. Understanding these connections is essential for developing a deep and holistic understanding of mathematics. In summary, solving the equation −x² − 4x + 5 = 0 has been a valuable learning experience. We have not only found the solutions but also deepened our understanding of quadratic equations and the principles of mathematical problem-solving. The solutions x’ = −5 and x” = 1 stand as a testament to our efforts and a symbol of the power of mathematics to unravel complex problems. The broader implications of understanding quadratic equations extend to numerous fields. In physics, quadratic equations are used to model projectile motion, the trajectory of objects thrown into the air. The solutions represent the times at which the object reaches a certain height or hits the ground. In engineering, quadratic equations are used to design structures, such as bridges and buildings, ensuring their stability and strength. In economics, quadratic equations are used to model supply and demand curves, helping to determine equilibrium prices and quantities. The applications of quadratic equations are vast and varied, underscoring their importance in the modern world. By mastering the techniques for solving these equations, we equip ourselves with the tools to tackle a wide range of real-world problems. Moreover, the problem-solving skills developed in the process of solving quadratic equations are transferable to other areas of life. The ability to break down a complex problem into smaller, manageable steps, to apply appropriate formulas and techniques, and to verify the solutions are valuable skills in any field. Whether we are working in science, technology, engineering, mathematics, or any other discipline, the principles of problem-solving learned through quadratic equations will serve us well. In conclusion, the solutions x’ = −5 and x” = 1 to the equation −x² − 4x + 5 = 0 are not just answers to a mathematical problem; they are symbols of the power of mathematics to illuminate the world around us. By understanding quadratic equations and their applications, we gain a deeper appreciation for the beauty and utility of mathematics.

Beyond the immediate solution of the equation −x² − 4x + 5 = 0, the principles and techniques used have far-reaching applications. Understanding quadratic equations is crucial for various fields, including physics, engineering, and economics. In physics, quadratic equations are fundamental in describing projectile motion, where the path of an object thrown into the air can be modeled by a parabola. The solutions to the quadratic equation represent the times at which the object reaches a certain height or hits the ground. This understanding is crucial for predicting the behavior of objects in motion and for designing systems that involve projectile motion, such as ballistics or sports equipment. In engineering, quadratic equations are used in structural design, electrical circuit analysis, and control systems. For example, the design of a bridge involves ensuring that it can withstand various loads and stresses. Quadratic equations can be used to model the forces acting on the bridge and to determine the optimal dimensions and materials to use. In electrical circuit analysis, quadratic equations can be used to calculate the current and voltage in circuits with resistors, capacitors, and inductors. In control systems, quadratic equations are used to design feedback loops that ensure the stability and performance of systems, such as robots or aircraft. The applications of quadratic equations in engineering are vast and diverse, highlighting their importance in the field. In economics, quadratic equations are used to model supply and demand curves, cost functions, and profit functions. For example, the equilibrium price and quantity in a market can be found by solving a system of equations that includes a quadratic equation. Cost functions, which describe the cost of producing a certain quantity of goods, can also be modeled using quadratic equations. Profit functions, which describe the profit earned from selling a certain quantity of goods, are often quadratic as well. Understanding these economic models is crucial for making informed business decisions and for analyzing the behavior of markets. The ability to apply quadratic equations to these diverse fields underscores their significance in scientific and economic analysis. The skills developed in solving quadratic equations, such as algebraic manipulation, problem-solving, and critical thinking, are transferable to a wide range of disciplines. Whether one is designing a bridge, analyzing a circuit, or modeling a market, the ability to think mathematically and to apply the principles of algebra is essential for success. The study of quadratic equations, therefore, is not just an academic exercise; it is a valuable preparation for a variety of careers and intellectual pursuits. Furthermore, the concepts underlying quadratic equations extend to more advanced mathematical topics. The quadratic formula, for example, is a special case of more general formulas for finding the roots of polynomials. The discriminant, which determines the nature of the solutions, is a concept that appears in many areas of mathematics, including number theory and abstract algebra. The techniques for solving quadratic equations, such as completing the square and factoring, are used in calculus and differential equations. By mastering quadratic equations, one lays a solid foundation for further study in mathematics and its applications. The historical development of quadratic equations also provides valuable insights into the evolution of mathematical thought. Quadratic equations were studied by ancient mathematicians, including the Babylonians and the Greeks. The Babylonians developed methods for solving quadratic equations using geometric techniques and numerical approximations. The Greeks, such as Euclid, used geometric constructions to represent and solve quadratic equations. The quadratic formula, as we know it today, was developed over centuries, with contributions from mathematicians in India, the Middle East, and Europe. The story of quadratic equations is a testament to the global and collaborative nature of mathematical discovery. In conclusion, the study of quadratic equations is a rich and rewarding experience. It not only provides us with a powerful tool for solving mathematical problems but also opens doors to a wide range of applications and intellectual pursuits. The solutions x’ = −5 and x” = 1 to the equation −x² − 4x + 5 = 0 are just the beginning of a journey into the fascinating world of mathematics and its applications.

When faced with a quadratic equation, such as −x² − 4x + 5 = 0, there are several methods available to find the solutions. These methods include factoring, completing the square, and the quadratic formula. Each method has its advantages and disadvantages, and the choice of method often depends on the specific equation and the solver's preferences. Factoring is a method that involves rewriting the quadratic equation as a product of two linear factors. For example, the equation −x² − 4x + 5 = 0 can be factored as −(x + 5)(x − 1) = 0. The solutions are then found by setting each factor equal to zero and solving for x. In this case, x + 5 = 0 gives x = −5, and x − 1 = 0 gives x = 1. Factoring is a quick and efficient method when it is applicable, but it is not always easy to find the factors, especially when the coefficients are large or the solutions are irrational. The advantage of factoring lies in its simplicity and directness. When the equation can be easily factored, it provides a straightforward path to the solutions. However, the limitation of factoring is that not all quadratic equations can be factored easily, or at all, using integer coefficients. This makes factoring less versatile than other methods, such as the quadratic formula. In addition, factoring requires a certain level of intuition and pattern recognition, which can be challenging for some students. The ability to quickly identify factorable equations is a skill that develops with practice, but it is not a guaranteed method for solving all quadratic equations. For these reasons, while factoring is a valuable technique to have in one's mathematical toolkit, it is often used in conjunction with other methods. Completing the square is a method that involves transforming the quadratic equation into a perfect square trinomial. This is done by adding and subtracting a constant term to both sides of the equation. For example, to complete the square for the equation −x² − 4x + 5 = 0, we first rewrite it as −(x² + 4x) + 5 = 0. Then, we add and subtract (4/2)² = 4 inside the parentheses: −(x² + 4x + 4 − 4) + 5 = 0. This gives −((x + 2)² − 4) + 5 = 0, which simplifies to −(x + 2)² + 9 = 0. The solutions are then found by isolating the squared term and taking the square root of both sides: (x + 2)² = 9, so x + 2 = ±3. This gives x = −2 ± 3, which yields the solutions x = 1 and x = −5. Completing the square is a more general method than factoring, as it can be applied to any quadratic equation. However, it is also more complex and time-consuming, especially when the coefficients are fractions or irrational numbers. The main advantage of completing the square is that it provides a systematic way to solve any quadratic equation. Unlike factoring, it does not rely on intuition or pattern recognition. This makes it a reliable method for students who may struggle with factoring. Another advantage of completing the square is that it can be used to derive the quadratic formula. This connection between completing the square and the quadratic formula provides a deeper understanding of the underlying mathematics. However, the complexity of the steps involved in completing the square can be a disadvantage, especially for equations with non-integer coefficients. The process can be tedious and prone to errors, which makes it less practical for quick problem-solving compared to the quadratic formula. The quadratic formula, as discussed earlier, is a universal solver for quadratic equations. It provides a direct method to find the solutions, regardless of the complexity of the equation. The quadratic formula is derived by completing the square on the general form of a quadratic equation, ax² + bx + c = 0. The advantage of the quadratic formula is its generality and ease of application. It can be used to solve any quadratic equation, regardless of whether it can be factored or whether the solutions are rational or irrational. The only requirement is to correctly identify the coefficients a, b, and c, and to substitute them into the formula. This makes the quadratic formula a highly reliable and efficient method. The main disadvantage of the quadratic formula is that it can be less intuitive than factoring or completing the square. It does not provide as much insight into the structure of the equation or the nature of the solutions. However, for most practical purposes, the quadratic formula is the method of choice, especially when speed and accuracy are paramount. In summary, each method has its strengths and weaknesses. Factoring is quick and efficient for equations that can be easily factored, but it is not always applicable. Completing the square is a general method that can be used to solve any quadratic equation, but it is more complex and time-consuming. The quadratic formula is a universal solver that provides a direct method to find the solutions, but it can be less intuitive. The best approach is to be familiar with all three methods and to choose the one that is most appropriate for the specific equation at hand. The choice of method also depends on the context of the problem. In some situations, such as in standardized tests, speed is of the essence, and the quadratic formula is the most efficient choice. In other situations, such as in a mathematical proof, completing the square may be preferred because it demonstrates a deeper understanding of the underlying concepts. Ultimately, the mastery of quadratic equations involves not only knowing how to solve them but also understanding the different methods available and when to use them. This flexibility and adaptability are the hallmarks of a skilled problem-solver.

Solutions

The correct answer is c) 𝑥’ = −5 𝑒 𝑥” = 1.

This article has provided a comprehensive guide to solving the quadratic equation −x² − 4x + 5 = 0, detailing the use of the quadratic formula and the verification of solutions. Understanding these steps is crucial for mastering quadratic equations and their applications in various fields.