Solving Quadratic Equations Finding The Other Root When One Root Is X=-2i
In the realm of mathematics, particularly when dealing with quadratic equations, understanding the nature of roots is crucial. Quadratic equations, characterized by the general form ax2 + bx + c = 0, where a, b, and c are constants and a ≠0, can have various types of solutions or roots. These roots can be real numbers, complex numbers, or a combination thereof. When one solution to a quadratic equation is a complex number, a fascinating property comes into play: the complex conjugate root theorem. This theorem provides a powerful shortcut for finding the other solution. This article delves into the intricacies of quadratic equations, complex roots, and the complex conjugate root theorem, offering a comprehensive guide to solving such problems. Let's explore the given problem where one solution to a quadratic equation is x = -2i, and our objective is to determine the other solution. This exploration will not only provide the answer but also enrich your understanding of the underlying mathematical principles. Understanding these principles is fundamental for success in algebra and beyond, as quadratic equations form the basis for many advanced mathematical concepts.
Complex Numbers and Quadratic Equations
Before we dive into the specific problem, it's essential to lay a solid foundation by understanding complex numbers and their relationship with quadratic equations. Complex numbers, denoted in the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1), extend the real number system. The term a represents the real part, and bi represents the imaginary part. When dealing with quadratic equations, complex roots arise when the discriminant (Δ = b2 - 4ac) is negative. The discriminant, a critical component of the quadratic formula, determines the nature of the roots. A negative discriminant signifies that the roots are complex conjugates, meaning they come in pairs of the form a + bi and a - bi. This pairing is not a coincidence; it's a direct consequence of the quadratic formula and the properties of complex numbers. The quadratic formula, given by x = (-b ± √(b2 - 4ac)) / (2a), clearly shows how the ± sign before the square root leads to two solutions. When the discriminant is negative, the square root of a negative number introduces the imaginary unit i, thus resulting in complex roots. These complex roots are not just abstract mathematical concepts; they have significant applications in various fields, including electrical engineering, quantum mechanics, and signal processing. Understanding their behavior and properties is therefore crucial for a well-rounded mathematical education. Furthermore, the complex conjugate root theorem, which we will discuss in detail later, provides a powerful tool for quickly identifying the other root when one complex root is known.
The Complex Conjugate Root Theorem
The cornerstone of solving our problem lies in the complex conjugate root theorem. This theorem states that if a polynomial equation with real coefficients has a complex root a + bi, then its complex conjugate a - bi is also a root. In simpler terms, complex roots of polynomial equations with real coefficients always occur in conjugate pairs. This theorem is not just a mathematical curiosity; it's a fundamental principle that arises from the properties of complex numbers and polynomial equations. To appreciate the theorem's significance, consider what happens when you substitute a complex number into a polynomial equation with real coefficients. The imaginary parts of the terms will combine in such a way that if a + bi is a root, then a - bi must also be a root to ensure the equation equals zero. The theorem drastically simplifies the process of finding roots. If you know one complex root, you immediately know its conjugate is also a root. This eliminates the need for lengthy calculations or the use of the quadratic formula. The theorem is particularly useful in higher-degree polynomial equations where finding roots can be quite challenging. By identifying one complex root, you can use the theorem to find another, effectively reducing the complexity of the problem. The complex conjugate root theorem is a powerful tool in algebra, and mastering it is essential for solving a wide range of polynomial equations, especially those with complex solutions. It's a testament to the elegant structure of mathematics, where seemingly abstract concepts like complex numbers follow predictable and useful patterns.
Applying the Theorem to Find the Other Solution
Now, let's apply the complex conjugate root theorem to our specific problem. We are given that one solution to a quadratic equation is x = -2i. This is a pure imaginary number, which can be expressed in the complex form as 0 - 2i. The complex conjugate of a number a + bi is a - bi. Therefore, the complex conjugate of -2i (or 0 - 2i) is 0 + 2i, which simplifies to 2i. According to the complex conjugate root theorem, if -2i is a solution, then its complex conjugate, 2i, must also be a solution. This elegantly solves our problem without requiring any complex calculations or the use of the quadratic formula. We have directly applied the theorem to identify the other root. This example perfectly illustrates the power and efficiency of the complex conjugate root theorem. It transforms a potentially complicated problem into a straightforward application of a fundamental principle. Understanding and remembering this theorem can save significant time and effort when solving quadratic equations with complex roots. It's a key concept that highlights the beauty and interconnectedness of mathematical ideas. Furthermore, this principle extends beyond quadratic equations and applies to polynomial equations of higher degrees, making it a versatile tool in the realm of algebra.
Step-by-Step Solution and Answer
To recap, we started with the information that one solution to a quadratic equation is x = -2i. We identified that this is a complex number in the form a + bi, where a = 0 and b = -2. We then invoked the complex conjugate root theorem, which states that if a polynomial equation with real coefficients has a complex root a + bi, its complex conjugate a - bi is also a root. Applying this theorem, we found the complex conjugate of -2i (which is 0 - 2i) to be 0 + 2i, or simply 2i. Therefore, the other solution to the quadratic equation is x = 2i. Now, let's look at the given options to select the correct answer:
- A. x = -9
- B. x = -3i
- C. x = 2i
- D. x = 2
Clearly, option C, x = 2i, matches our solution. Thus, the correct answer is C. This step-by-step approach not only provides the answer but also reinforces the process of applying the complex conjugate root theorem. By carefully following each step, you can confidently solve similar problems and deepen your understanding of complex roots and quadratic equations. This methodical approach is crucial for success in mathematics, as it allows you to break down complex problems into manageable steps and apply the appropriate theorems and techniques. Furthermore, practicing this approach will enhance your problem-solving skills and build a solid foundation for more advanced mathematical concepts.
Conclusion: Mastering Complex Roots
In conclusion, this exploration of quadratic equations and complex roots has demonstrated the power and elegance of the complex conjugate root theorem. We successfully found the other solution to the quadratic equation by applying this theorem to the given root x = -2i. The correct answer is C. x = 2i. Understanding the complex conjugate root theorem is essential for anyone studying algebra and higher-level mathematics. It provides a shortcut for finding complex roots, eliminating the need for tedious calculations. Moreover, it underscores the interconnectedness of mathematical concepts, showing how complex numbers and polynomial equations are related. By mastering this theorem, you gain a valuable tool for solving a wide range of problems and deepen your appreciation for the beauty and structure of mathematics. The ability to confidently work with complex roots is not just an academic exercise; it has practical applications in various fields, including engineering, physics, and computer science. So, embrace the power of complex numbers and the complex conjugate root theorem, and you'll be well-equipped to tackle any quadratic equation that comes your way. Continue practicing and exploring these concepts, and you'll find yourself becoming more proficient and confident in your mathematical abilities.
