Solving $3x^2 + 6x + 15 = 0$ A Comprehensive Guide

Introduction

In this comprehensive article, we will delve into the process of solving the quadratic equation $3x^2 + 6x + 15 = 0$. Quadratic equations, which are polynomial equations of the second degree, play a crucial role in various fields such as mathematics, physics, engineering, and computer science. Understanding how to solve them is a fundamental skill for anyone working with these disciplines. This exploration will meticulously walk through the steps required to find the solutions, highlighting the underlying concepts and techniques involved. This detailed approach ensures that both novice learners and experienced practitioners can grasp the methodology effectively and apply it to similar problems.

Quadratic equations are generally expressed in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ represents the unknown variable. The solutions to these equations, also known as roots, can be real or complex numbers. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. In this case, we will primarily focus on using the quadratic formula to find the solutions for the given equation. The quadratic formula is a powerful tool that provides a direct method for finding the roots of any quadratic equation, regardless of its complexity. By understanding and applying this formula, one can efficiently solve a wide range of problems involving quadratic equations. The versatility and reliability of the quadratic formula make it an indispensable technique in mathematical problem-solving.

The significance of mastering quadratic equations extends beyond academic exercises. In practical applications, these equations are used to model a variety of phenomena, from projectile motion in physics to optimization problems in engineering. For instance, determining the trajectory of a projectile under the influence of gravity often involves solving a quadratic equation. Similarly, optimizing the design of a structure or a system may require finding the roots of a quadratic function. The ability to solve quadratic equations is, therefore, not just a theoretical exercise but a practical necessity in many professional fields. This article aims to equip readers with the necessary skills and understanding to tackle such real-world problems with confidence.

Method 1: Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by:

x = rac{-b \&pm \&sqrt{b^2 - 4ac}}{2a}

In our equation, $3x^2 + 6x + 15 = 0$, we can identify the coefficients as follows:

• $a = 3$
• $b = 6$
• $c = 15$

Now, we substitute these values into the quadratic formula:

x = rac{-6 \&pm \&sqrt{6^2 - 4(3)(15)}}{2(3)}

Let's simplify the expression step by step. First, calculate the value inside the square root:

$$6^2 - 4(3)(15) = 36 - 180 = -144$$

Now, substitute this value back into the formula:

x = rac{-6 \&pm \&sqrt{-144}}{6}

Since the discriminant (the value inside the square root) is negative, we will have complex solutions. Recall that $\&sqrt{-1} = i$, where $i$ is the imaginary unit. So, $\&sqrt{-144} = \&sqrt{144} \&times \&sqrt{-1} = 12i$. Substitute this back into the equation:

x = rac{-6 \&pm \&12i}{6}

Now, divide both the real and imaginary parts by 6:

$$x = -1 \&pm \&2i$$

Thus, the solutions are $x = -1 + 2i$ and $x = -1 - 2i$.

The quadratic formula's elegance lies in its ability to handle any quadratic equation, regardless of the nature of its roots. Whether the roots are real and distinct, real and repeated, or complex conjugates, the formula provides a straightforward and reliable method for finding them. This universality makes it an indispensable tool in mathematics and related fields. The process of substituting coefficients into the formula and simplifying the resulting expression may seem mechanical at first, but with practice, it becomes second nature. This mastery is crucial for tackling more advanced mathematical concepts that build upon the foundation of quadratic equations. Furthermore, the quadratic formula serves as a cornerstone for understanding other numerical methods used to solve polynomial equations of higher degrees.

Understanding the implications of the discriminant, $b^2 - 4ac$, is also essential. A positive discriminant indicates two distinct real roots, a zero discriminant indicates a repeated real root, and a negative discriminant indicates two complex conjugate roots. This insight allows one to predict the nature of the solutions even before applying the quadratic formula. In our example, the negative discriminant clearly signaled the presence of complex solutions, which were subsequently confirmed through the calculations. The ability to interpret the discriminant enhances problem-solving efficiency and provides a deeper understanding of the relationship between the coefficients of a quadratic equation and its roots.

Method 2: Completing the Square (Alternative Method)

Completing the square is another method to solve quadratic equations. While the quadratic formula is often more direct, completing the square provides a deeper understanding of the structure of quadratic equations and can be useful in other contexts, such as deriving the quadratic formula itself. To solve $3x^2 + 6x + 15 = 0$ by completing the square, we first divide the entire equation by the leading coefficient, 3:

$$x^2 + 2x + 5 = 0$$

Next, we want to rewrite the left side in the form $(x + h)^2 + k$, where $h$ and $k$ are constants. To do this, we take half of the coefficient of the $x$ term (which is 2), square it (which gives 1), and add and subtract it within the equation:

$$x^2 + 2x + 1 - 1 + 5 = 0$$

Now, we can group the first three terms to form a perfect square:

$$(x + 1)^2 + 4 = 0$$

Subtract 4 from both sides:

$$(x + 1)^2 = -4$$

Take the square root of both sides:

$$x + 1 = \&pm \&sqrt{-4}$$

Since $\&sqrt{-4} = 2i$, we have:

$$x + 1 = \&pm \&2i$$

Finally, subtract 1 from both sides to solve for $x$:

$$x = -1 \&pm \&2i$$

As we can see, this method yields the same solutions as the quadratic formula: $x = -1 + 2i$ and $x = -1 - 2i$.

Completing the square's value extends beyond simply finding the solutions to quadratic equations. It is a fundamental technique used in various areas of mathematics, including calculus, where it is essential for integrating certain types of functions. The method involves transforming a quadratic expression into a perfect square trinomial plus a constant, which can then be easily manipulated. This transformation not only simplifies the process of solving the equation but also provides valuable insights into the structure and properties of quadratic functions. For example, by completing the square, one can easily determine the vertex of a parabola, which is the minimum or maximum point of the function.

Furthermore, the process of completing the square is closely related to the derivation of the quadratic formula. Understanding the steps involved in completing the square provides a deeper appreciation of the quadratic formula and its underlying principles. This connection underscores the importance of mastering both techniques, as they complement each other and provide a more comprehensive understanding of quadratic equations. In addition to its theoretical significance, completing the square has practical applications in various fields, such as optimization problems, where it can be used to find the maximum or minimum value of a quadratic function subject to certain constraints. The method's versatility and widespread applicability make it an indispensable tool in the mathematician's toolkit.

Conclusion

In conclusion, we have successfully solved the quadratic equation $3x^2 + 6x + 15 = 0$ using both the quadratic formula and completing the square. Both methods yielded the same complex solutions, $x = -1 + 2i$ and $x = -1 - 2i$. The correct answer is therefore B. $-1 \&pm 2i$. This exercise demonstrates the effectiveness of these methods in handling quadratic equations with complex roots. Mastering these techniques is essential for anyone studying mathematics and related fields, as quadratic equations frequently appear in various applications and theoretical problems.

The importance of understanding different methods for solving quadratic equations cannot be overstated. While the quadratic formula provides a direct and efficient approach, completing the square offers a deeper understanding of the underlying structure of quadratic functions. This dual perspective enhances problem-solving skills and provides a more comprehensive mathematical foundation. Furthermore, proficiency in these techniques is crucial for tackling more advanced mathematical concepts, such as calculus and differential equations, where quadratic equations often play a key role. The ability to solve quadratic equations accurately and efficiently is, therefore, a fundamental skill for success in many scientific and engineering disciplines.

Moreover, the solutions to quadratic equations have significant practical applications in various fields. For example, in physics, they are used to model projectile motion and determine the trajectory of objects under the influence of gravity. In engineering, they are used in the design of structures and systems, where optimization problems often involve solving quadratic equations. In computer science, they are used in algorithms for optimization and machine learning. The wide range of applications underscores the importance of mastering the techniques discussed in this article. By understanding the theory and practice of solving quadratic equations, students and professionals can confidently tackle a wide range of real-world problems and make valuable contributions to their respective fields.