Solving 3x² + 6x + 10 = 0 A Step-by-Step Guide To Complex Solutions

Let's dive into solving the quadratic equation 3x² + 6x + 10 = 0. Quadratic equations are a fundamental part of algebra, and this particular equation leads us to the fascinating world of complex numbers. So, buckle up, guys, because we're about to embark on a mathematical adventure!

Understanding Quadratic Equations

Before we jump into the solution, let's quickly recap what quadratic equations are. A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The solutions to a quadratic equation are also known as its roots. These roots are the values of x that satisfy the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. The quadratic formula is a universally applicable method that works for any quadratic equation, regardless of whether its roots are real or complex. In our case, we'll primarily use the quadratic formula because it’s the most straightforward approach for this equation. However, understanding the other methods can provide a broader perspective on problem-solving in algebra. Factoring is typically the quickest method when the equation can be easily factored, but it's not always feasible. Completing the square is another powerful technique that can be used not only to solve quadratic equations but also to derive the quadratic formula itself. Each method has its own advantages and situations where it's most effective, making a comprehensive understanding of these techniques essential for anyone studying algebra. Remember, the goal is not just to find the solution but also to understand the underlying principles that make the solution possible. This deeper understanding allows us to tackle more complex problems and apply these concepts in various mathematical contexts.

Identifying Coefficients

The first step in solving our equation, 3x² + 6x + 10 = 0, using the quadratic formula is to identify the coefficients a, b, and c. In our equation:

• a = 3 (the coefficient of x²)
• b = 6 (the coefficient of x)
• c = 10 (the constant term)

Identifying these coefficients correctly is crucial because they will be plugged directly into the quadratic formula. A mistake here can lead to an incorrect solution, so double-checking is always a good idea. These coefficients dictate the shape and position of the parabola represented by the quadratic equation, and they influence the nature of the roots. For instance, the sign of 'a' determines whether the parabola opens upwards (if 'a' is positive) or downwards (if 'a' is negative). The value of 'b' affects the axis of symmetry, and 'c' represents the y-intercept of the parabola. By recognizing these connections, we can gain a deeper understanding of how each coefficient contributes to the overall behavior of the quadratic equation. This understanding is not just limited to solving equations; it extends to graphing and analyzing quadratic functions as well. Furthermore, identifying coefficients is a fundamental skill that extends to higher-degree polynomials and other mathematical contexts, emphasizing its importance in mathematical problem-solving. So, making sure we get these right is not just about solving this equation but also about building a solid foundation for future mathematical endeavors.

Applying the Quadratic Formula

The quadratic formula is a powerful tool that gives us the solutions (roots) of any quadratic equation in the form ax² + bx + c = 0. The formula is:

x = (-b ± √(b² - 4ac)) / 2a

Now, let's plug in our coefficients a = 3, b = 6, and c = 10 into the formula:

x = (-6 ± √(6² - 4 * 3 * 10)) / (2 * 3)

Simplifying this step by step is important to avoid errors. First, we calculate the discriminant, which is the term under the square root, b² - 4ac. The discriminant tells us a lot about the nature of the roots. If it's positive, we have two distinct real roots; if it's zero, we have one real root (a repeated root); and if it's negative, we have two complex roots. This is a critical concept, so make sure you understand it well. The quadratic formula is derived by completing the square on the general form of the quadratic equation, so understanding its derivation can provide further insight into why it works. Moreover, the quadratic formula is not just a standalone formula; it's a cornerstone of algebra that connects various concepts. It is also used in various fields of science and engineering. By mastering the quadratic formula, we not only solve equations but also deepen our appreciation for the interconnectedness of mathematical ideas. Remember, practice makes perfect, so try solving various quadratic equations to become comfortable with the application of the formula. This will help you recognize patterns and develop problem-solving strategies that will be useful in more advanced mathematical studies.

Simplifying the Expression

Let's simplify the expression we obtained after applying the quadratic formula:

x = (-6 ± √(36 - 120)) / 6

x = (-6 ± √(-84)) / 6

Notice that we have a negative number under the square root. This indicates that the solutions will be complex numbers. Complex numbers involve the imaginary unit i, where i² = -1. Dealing with a negative number under the square root might seem daunting at first, but it's a standard situation when solving quadratic equations, especially when the discriminant (the part under the square root) is negative. Embracing this concept opens the door to a whole new world of mathematical possibilities. Simplifying the expression involves several steps, each requiring careful attention to detail. First, we need to express √(-84) in terms of i. To do this, we factor out -1 and find the largest perfect square that divides 84. Understanding how to manipulate square roots and complex numbers is essential for success in algebra and beyond. The ability to simplify complex expressions is not just a skill for solving equations; it's a fundamental tool in many areas of mathematics, physics, and engineering. So, taking the time to master these techniques will pay dividends in your mathematical journey. Remember, the key is to break down the problem into smaller, manageable steps and to understand the reasoning behind each step. This approach will not only help you solve the problem at hand but also build your confidence and problem-solving skills in general.

Introducing Complex Numbers

Since we have √(-84), we need to express it using the imaginary unit i. We can rewrite √(-84) as √(84 * -1) = √84 * √-1 = √84 * i. Now, we simplify √84. The prime factorization of 84 is 2 * 2 * 3 * 7, so √84 = √(2² * 21) = 2√21. Therefore, √(-84) = 2√21 * i.

This step is pivotal in understanding how to handle complex solutions. Complex numbers are numbers that have a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part. When we encounter a negative number under a square root, it signals that we're dealing with complex roots. Embracing complex numbers expands the landscape of mathematical solutions, allowing us to solve equations that would otherwise be unsolvable within the realm of real numbers. The introduction of the imaginary unit i might seem like an abstract concept, but it's a powerful tool that allows us to express and manipulate these complex solutions. Understanding how to work with complex numbers is not just important for solving quadratic equations; it's crucial for many areas of mathematics, physics, and engineering. Complex numbers are used in electrical engineering, quantum mechanics, and signal processing, to name just a few fields. So, grasping the fundamentals of complex numbers is a valuable investment in your mathematical education.

Finding the Complex Solutions

Now we substitute 2√21 * i back into our equation:

x = (-6 ± 2√21 * i) / 6

We can simplify this further by dividing both the real and imaginary parts by 6:

x = -1 ± (√21 / 3)i

So, the two complex solutions are:

• x₁ = -1 + (√21 / 3)*i
• x₂ = -1 - (√21 / 3)*i

These are the complex roots of the quadratic equation 3x² + 6x + 10 = 0. The solutions are complex conjugates, meaning they have the same real part but opposite imaginary parts. This is a common characteristic of quadratic equations with real coefficients and a negative discriminant. Identifying this pattern can help you anticipate the nature of the solutions and double-check your work. The process of simplifying the complex solutions involves a few algebraic manipulations, including dividing both the real and imaginary parts by the denominator. This step is crucial to express the solutions in their simplest form. The final solutions, x₁ and x₂, represent two points in the complex plane. Complex numbers can be visualized as points on a two-dimensional plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This geometric interpretation adds another layer of understanding to complex numbers and their solutions. Recognizing that complex solutions come in conjugate pairs and understanding how to simplify and express them correctly are key takeaways from this exercise. This knowledge will serve you well in future mathematical endeavors.

Conclusion

So, there you have it! We've successfully solved the quadratic equation 3x² + 6x + 10 = 0 and found its complex solutions. This exercise demonstrates the power of the quadratic formula and the importance of understanding complex numbers. Remember, math isn't just about finding the right answer; it's about the journey of understanding the concepts and the methods. Keep practicing, keep exploring, and you'll conquer any mathematical challenge that comes your way, guys! Solving quadratic equations with complex solutions might seem tricky at first, but with a solid understanding of the quadratic formula, complex numbers, and algebraic manipulation, it becomes a manageable and even enjoyable task. The key is to break down the problem into smaller steps, pay attention to detail, and practice regularly. Each time you solve a similar problem, you reinforce your understanding and build your confidence. Moreover, remember that mathematics is a cumulative subject. The concepts you learn in one area often build upon and connect to concepts in other areas. So, the skills you've developed in solving quadratic equations will be valuable in other mathematical contexts, such as calculus, differential equations, and linear algebra. The journey through mathematics is a continuous learning process, and every problem you solve contributes to your overall mathematical maturity. Keep an open mind, embrace the challenges, and enjoy the beauty and elegance of mathematics!