Solving K^2 + 7k + 19 = 0 A Step By Step Guide

Introduction

In this article, we will delve into the process of solving the quadratic equation k^2 + 7k + 19 = 0. Quadratic equations, characterized by their second-degree polynomial form, frequently appear in various branches of mathematics and physics. Finding the solutions, also known as roots or zeros, of a quadratic equation is a fundamental skill. We will employ the quadratic formula, a powerful tool for finding these solutions, even when they are non-real (complex numbers). This article will guide you through each step, ensuring a clear understanding of the process and the underlying concepts. Our focus will be on achieving a fully simplified answer, including both real and non-real solutions. Understanding how to solve quadratic equations is crucial for further studies in mathematics and related fields. The quadratic formula allows us to solve any quadratic equation, regardless of whether it can be factored easily or not. Moreover, recognizing and dealing with non-real solutions expands our mathematical toolkit and prepares us for more advanced topics. The equation k^2 + 7k + 19 = 0 serves as an excellent example to demonstrate the application of the quadratic formula and the nature of complex roots. So, let's embark on this mathematical journey and unlock the solutions to this intriguing equation.

Understanding Quadratic Equations

Before diving into the solution, let's first establish a solid understanding of quadratic equations. A quadratic equation is a polynomial equation of the second degree, generally expressed in the standard form ax^2 + bx + c = 0, where 'a', 'b', and 'c' are coefficients, and 'x' represents the variable. The coefficient 'a' cannot be zero, otherwise, the equation would reduce to a linear equation. The solutions to a quadratic equation, often referred to as roots or zeros, are the values of 'x' that satisfy the equation. These roots can be real numbers, complex numbers, or a combination of both. The nature of the roots is determined by the discriminant, which we will discuss later. Quadratic equations have a rich history and are used extensively in various applications, from calculating trajectories in physics to modeling growth and decay in finance. They also form the basis for more advanced algebraic concepts. Understanding the properties and solutions of quadratic equations is therefore essential for anyone pursuing further studies in science, engineering, or mathematics. In our specific equation, k^2 + 7k + 19 = 0, we can identify 'a' as 1, 'b' as 7, and 'c' as 19. Recognizing these coefficients is the first step towards applying the quadratic formula. Mastering the concepts of quadratic equations not only helps in solving mathematical problems but also enhances analytical and problem-solving skills, which are valuable in many aspects of life. With a firm grasp of the fundamentals, we can confidently proceed to the solution process.

The Quadratic Formula: Our Key Tool

The cornerstone of solving quadratic equations is the quadratic formula. This formula provides a direct method for finding the roots of any quadratic equation in the form ax^2 + bx + c = 0. The formula is expressed as:

x = (-b ± √(b^2 - 4ac)) / (2a)

Where:

• 'x' represents the roots of the quadratic equation.
• 'a', 'b', and 'c' are the coefficients of the quadratic equation.

The quadratic formula is derived by completing the square on the general form of the quadratic equation. It is a versatile tool because it works for all quadratic equations, regardless of whether they can be easily factored or not. The expression inside the square root, b^2 - 4ac, is known as the discriminant. The discriminant provides valuable information about 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). If the discriminant is negative, the equation has two complex conjugate roots. Understanding the quadratic formula and the role of the discriminant is crucial for solving quadratic equations efficiently and accurately. In our specific case, k^2 + 7k + 19 = 0, we will substitute the values of 'a', 'b', and 'c' into the quadratic formula to find the solutions for 'k'. The quadratic formula not only gives us the solutions but also provides insights into the nature of these solutions, which is essential for a complete understanding of the equation. With the quadratic formula at our disposal, we are well-equipped to tackle the equation and unveil its roots.

Applying the Quadratic Formula to k^2 + 7k + 19 = 0

Now, let's put the quadratic formula into action to solve our equation: k^2 + 7k + 19 = 0. As we identified earlier, the coefficients are:

• a = 1
• b = 7
• c = 19

Substituting these values into the quadratic formula, we get:

k = (-7 ± √(7^2 - 4 * 1 * 19)) / (2 * 1)

Next, we simplify the expression step-by-step. First, we calculate the square of 'b' and the product of '4ac':

k = (-7 ± √(49 - 76)) / 2

Now, we subtract 76 from 49:

k = (-7 ± √(-27)) / 2

Notice that the value inside the square root is negative. This indicates that the solutions will be complex numbers. The square root of -27 can be expressed as √(-1 * 27), which is equal to √(-1) * √(27). We know that √(-1) is defined as the imaginary unit 'i', and √(27) can be simplified to 3√3. Therefore, we have:

k = (-7 ± 3i√3) / 2

This is the simplified form of the solutions. We have successfully applied the quadratic formula and simplified the result, revealing the complex roots of the equation. The process of substituting the coefficients, simplifying the expression, and dealing with the negative square root is crucial for mastering the application of the quadratic formula. The solutions we obtained are in the form of complex numbers, which demonstrates the versatility of the quadratic formula in handling different types of roots. With these solutions, we have a complete understanding of the equation's behavior and its roots.

Simplifying the Solutions: Dealing with Complex Numbers

As we found in the previous section, the solutions to the equation k^2 + 7k + 19 = 0 are complex numbers. Let's further simplify these solutions and express them in the standard form of a complex number, which is a + bi, where 'a' is the real part and 'b' is the imaginary part. Our solutions are:

k = (-7 ± 3i√3) / 2

We can separate this into two distinct solutions:

k1 = (-7 + 3i√3) / 2

k2 = (-7 - 3i√3) / 2

Now, we can rewrite each solution in the standard complex number form by dividing both the real and imaginary parts by 2:

k1 = -7/2 + (3√3 / 2)i

k2 = -7/2 - (3√3 / 2)i

These are the fully simplified complex solutions to the quadratic equation. Notice that the solutions are complex conjugates of each other, meaning they have the same real part but opposite imaginary parts. This is a common characteristic of quadratic equations with a negative discriminant. Understanding how to simplify complex numbers is essential for working with quadratic equations that have non-real roots. The process involves identifying the real and imaginary parts and expressing the solutions in the standard form. These simplified solutions provide a clear representation of the roots and their relationship to each other. By expressing the solutions in the standard complex number form, we gain a deeper understanding of their properties and can easily use them in further calculations or applications. We have now successfully simplified the solutions, completing the process of solving the quadratic equation.

Verification of the Solutions

To ensure the accuracy of our solutions, it's crucial to verify them by substituting them back into the original equation, k^2 + 7k + 19 = 0. This process will confirm that the values we obtained for 'k' indeed satisfy the equation. We have two complex solutions:

k1 = -7/2 + (3√3 / 2)i

k2 = -7/2 - (3√3 / 2)i

Let's substitute k1 into the equation:

(-7/2 + (3√3 / 2)i)^2 + 7(-7/2 + (3√3 / 2)i) + 19 = 0

Expanding the terms, we get:

(49/4 - (21√3 / 2)i - 27/4) + (-49/2 + (21√3 / 2)i) + 19 = 0_

Simplifying further:

(22/4 - (21√3 / 2)i) + (-49/2 + (21√3 / 2)i) + 19 = 0_

(11/2 - (21√3 / 2)i) + (-49/2 + (21√3 / 2)i) + 19 = 0_

Combining the real and imaginary parts:

(11/2 - 49/2 + 19) + (- (21√3 / 2)i + (21√3 / 2)i) = 0

(-38/2 + 19) + 0i = 0

-19 + 19 = 0

0 = 0

The equation holds true for k1. Similarly, substituting k2 into the equation will also result in 0 = 0, confirming that k2 is also a valid solution. This verification process is a vital step in solving any equation, as it ensures that the obtained solutions are accurate and reliable. By substituting the solutions back into the original equation and confirming that the equation is satisfied, we gain confidence in our results. This step not only validates our calculations but also reinforces our understanding of the equation and its solutions. The successful verification of both solutions demonstrates the correctness of our application of the quadratic formula and the simplification process. With this verification, we can confidently conclude that we have accurately solved the quadratic equation.

Conclusion

In this comprehensive guide, we successfully solved the quadratic equation k^2 + 7k + 19 = 0. We began by understanding the basics of quadratic equations and the importance of the quadratic formula. We then applied the quadratic formula to our specific equation, carefully substituting the coefficients and simplifying the expression. The solutions we obtained were complex numbers, which we further simplified and expressed in the standard complex number form. Finally, we verified our solutions by substituting them back into the original equation, confirming their accuracy. This process highlights the importance of each step in solving quadratic equations, from understanding the fundamental concepts to verifying the final results. The quadratic formula is a powerful tool that allows us to solve any quadratic equation, regardless of the nature of its roots. Mastering this formula and the techniques for simplifying complex numbers is essential for anyone studying mathematics or related fields. The ability to solve quadratic equations is a fundamental skill that has numerous applications in various disciplines. By working through this example, we have not only found the solutions to a specific equation but also reinforced our understanding of the general principles and techniques involved in solving quadratic equations. This knowledge will serve as a solid foundation for tackling more complex mathematical problems in the future. The systematic approach we followed, including understanding the equation, applying the appropriate formula, simplifying the results, and verifying the solutions, is a valuable strategy that can be applied to many other mathematical problems.