Solving X² + 5x + 7 = 0 A Step By Step Guide

Introduction: Understanding Quadratic Equations

In the realm of mathematics, quadratic equations hold a pivotal role, appearing in various contexts from basic algebra to advanced calculus and physics. A quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form of ax² + bx + c = 0, where a, b, and c are constants, and x represents the unknown variable. The solutions to these equations, also known as roots or zeros, are the values of x that satisfy the equation. Finding these roots is a fundamental skill in algebra, and several methods exist to accomplish this, each with its own strengths and applications. In this comprehensive guide, we will delve into the process of solving the specific quadratic equation x² + 5x + 7 = 0, exploring the quadratic formula, its derivation, and its application in detail. This exploration will not only provide the solution to this particular equation but also equip you with a robust understanding of how to tackle similar problems in the future. Remember, mastering quadratic equations opens doors to a deeper understanding of mathematical concepts and their applications in real-world scenarios, making it an essential skill for students, engineers, scientists, and anyone with a passion for problem-solving. Understanding the nature of quadratic equations and the techniques used to solve them is crucial for various applications in science, engineering, and economics. This article provides a step-by-step solution to the equation x² + 5x + 7 = 0 using the quadratic formula.

The Quadratic Formula: A Powerful Tool

The quadratic formula is a universally applicable method for solving quadratic equations of the form ax² + bx + c = 0. This formula provides a direct way to find the roots, regardless of whether the equation can be easily factored or not. The quadratic formula is expressed as:

x = (-b ± √(b² - 4ac)) / 2a

This formula is derived from the method of completing the square, a technique that transforms the quadratic equation into a perfect square trinomial, making it easier to solve. The quadratic formula is a powerful tool because it can be used to solve any quadratic equation, regardless of the nature of its roots. The discriminant, b² - 4ac, 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. Understanding the discriminant is essential for interpreting the solutions obtained from the quadratic formula. The beauty of the quadratic formula lies in its ability to handle any quadratic equation, making it an indispensable tool in algebra. While other methods, such as factoring, may be quicker for certain equations, the quadratic formula provides a consistent and reliable approach for all cases. Moreover, the quadratic formula provides insights into the nature of the roots themselves. By examining the discriminant (the part of the formula under the square root), we can determine whether the roots are real or complex, distinct or repeated, without even fully solving the equation. This ability to predict the nature of the roots adds another layer of understanding to the solution process and highlights the importance of the quadratic formula in the study of quadratic equations.

Applying the Quadratic Formula to x² + 5x + 7 = 0

To solve the given equation, x² + 5x + 7 = 0, we first identify the coefficients a, b, and c. In this case:

• a = 1
• b = 5
• c = 7

Now, we substitute these values into the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. This substitution is a critical step in applying the quadratic formula, as it ensures that the correct values are used in the subsequent calculations. Care must be taken to accurately identify the coefficients and substitute them into the formula, paying close attention to the signs. A common mistake is to misidentify the signs of the coefficients, which can lead to incorrect solutions. By carefully substituting the values, we set the stage for a straightforward calculation of the roots of the equation. The process of substituting the coefficients into the quadratic formula is a mechanical one, but it requires precision and attention to detail. Once the values are correctly substituted, the next step involves simplifying the expression, which will lead us to the solutions of the equation. This step-by-step approach ensures that we follow a systematic procedure, minimizing the chances of errors and maximizing our understanding of the process. The accurate substitution of coefficients is the foundation for solving quadratic equations using the quadratic formula, and it is a skill that is essential for any student of algebra.

Substituting the values, we get:

x = (-5 ± √(5² - 4 * 1 * 7)) / (2 * 1)

Simplifying the Expression

Next, we simplify the expression step by step. First, we calculate the value inside the square root:

5² - 4 * 1 * 7 = 25 - 28 = -3

This leads us to:

x = (-5 ± √(-3)) / 2

The negative value under the square root indicates that the roots are complex numbers. Complex numbers involve the imaginary unit i, where i² = -1. Thus, we can rewrite √(-3) as √3 * i. Simplifying the expression under the square root is a crucial step in solving quadratic equations using the quadratic formula. It allows us to determine the nature of the roots and express them in their simplest form. In this case, the negative value under the square root reveals that the roots are complex, which means they have both a real and an imaginary part. Understanding how to simplify radicals, especially those involving negative numbers, is essential for working with complex numbers and solving quadratic equations that have complex roots. The ability to manipulate and simplify expressions involving radicals is a fundamental skill in algebra and is used extensively in various branches of mathematics and science. This step of simplification not only brings us closer to the final solution but also deepens our understanding of the underlying mathematical concepts involved in solving quadratic equations.

Expressing the Complex Roots

Now, we express the solution in terms of complex numbers:

x = (-5 ± √3 * i) / 2

This can be written as two separate solutions:

x₁ = (-5 + √3 * i) / 2

x₂ = (-5 - √3 * i) / 2

These are the two complex roots of the equation x² + 5x + 7 = 0. Expressing the solutions in terms of complex numbers is the final step in solving quadratic equations that have a negative discriminant. Complex numbers are a fundamental concept in mathematics and are used extensively in various fields, including electrical engineering, quantum mechanics, and signal processing. Understanding how to work with complex numbers is essential for solving a wide range of mathematical problems. In this case, the complex roots of the equation x² + 5x + 7 = 0 provide a complete solution to the problem and demonstrate the power of the quadratic formula in handling all types of quadratic equations, regardless of the nature of their roots. The ability to express solutions in complex form expands our mathematical toolkit and allows us to solve problems that would otherwise be intractable. This final step of expressing the complex roots highlights the importance of a comprehensive understanding of mathematical concepts and their applications.

Conclusion: The Solution and Its Significance

Therefore, the solutions to the equation x² + 5x + 7 = 0 are:

x = (-5 ± √3 * i) / 2

Comparing this result with the given options, the correct answer is closest to option D, which is x = (-5 ± √(-3)) / 2. This comprehensive solution demonstrates the application of the quadratic formula in finding the roots of a quadratic equation, especially when the roots are complex. The quadratic formula is a fundamental tool in algebra, allowing us to solve any quadratic equation, regardless of the nature of its roots. Understanding the quadratic formula and its application is crucial for success in mathematics and related fields. This solution not only provides the answer to the specific equation but also reinforces the broader concepts of quadratic equations, complex numbers, and the importance of systematic problem-solving. By mastering these concepts, students and practitioners can confidently tackle a wide range of mathematical challenges and apply their knowledge to real-world problems. The significance of this solution lies not only in the answer itself but also in the process of obtaining it, which highlights the power of mathematical tools and the importance of a thorough understanding of fundamental concepts. In summary, the solution to the equation x² + 5x + 7 = 0 is a testament to the elegance and utility of the quadratic formula in solving quadratic equations, even when the roots are complex.

This step-by-step approach ensures clarity and accuracy in solving the quadratic equation.