How To Find Zeros Of F(x) = X^4 - 4x^2 - 5: A Step-by-Step Guide

Hey everyone! Let's dive into finding the zeros of the function $f(x) = x^4 - 4x^2 - 5$. Finding the zeros of a function, also known as finding the roots, means figuring out the x-values where the function equals zero. Basically, we're looking for the points where the graph of the function crosses the x-axis. This is a fundamental concept in algebra, and it's super useful in various fields like engineering, physics, and computer science. The function we're dealing with here is a quartic function (degree 4), which means it might have up to four real or complex roots. Don't worry, it's not as scary as it sounds! We can solve this using a clever substitution that turns it into a quadratic equation, which we all know how to handle. Remember, the zeros of a function are the solutions to the equation $f(x) = 0$. So, our goal is to solve $x^4 - 4x^2 - 5 = 0$. Let's break this down step-by-step to make sure everyone understands. It's all about making the complex simple, guys. We'll walk through the process, explaining each step, so you can apply this technique to other similar problems. By the end of this, you'll be a pro at finding zeros of these types of functions! Understanding how to solve for zeros is essential for analyzing the behavior of functions. It helps you identify where the function changes sign, which is crucial for graphing and understanding the function's overall shape. So, let's get started and make this a piece of cake! We'll first introduce a substitution to simplify the equation, solve the resulting quadratic equation, and finally, find the values of x. It's all about transforming the problem into something we can easily manage. Let's get to it! This is gonna be fun, and you'll be a math whiz in no time!

Step-by-Step Solution to Find the Zeros

Alright, let's get our hands dirty and find those zeros! The key here is to notice that the equation $x^4 - 4x^2 - 5 = 0$ has a special structure. Notice that the powers of x are decreasing, and they're all even. This hints at a clever substitution. The most crucial part is to recognize the pattern and use it to simplify things. First, we'll make a substitution. Let $y = x^2$. This means wherever we see an $x^2$ in our original equation, we're going to replace it with $y$. Our equation then becomes: $y^2 - 4y - 5 = 0$. This, my friends, is a quadratic equation! Much easier to handle than the quartic we started with, right? We can now solve this new quadratic equation for $y$. There are a couple of ways to do this: we can factor it, complete the square, or use the quadratic formula. Factoring is often the easiest method when it works, so let's try that first. Can we find two numbers that multiply to -5 and add up to -4? Yes! Those numbers are -5 and 1. So, we can factor the quadratic equation as $(y - 5)(y + 1) = 0$. Now, we set each factor equal to zero and solve for $y$. For the first factor, $y - 5 = 0$, which gives us $y = 5$. For the second factor, $y + 1 = 0$, which gives us $y = -1$. Now, remember that we initially substituted $y = x^2$. We're not done yet because we need to find the values of $x$. So, we'll substitute back to find the x-values. For $y = 5$, we have $x^2 = 5$. Taking the square root of both sides, we get $x = extbf{±}\\sqrt{5}$. For $y = -1$, we have $x^2 = -1$. Taking the square root of both sides, we get $x = extbf{±}i$, where $i$ is the imaginary unit, defined as $i = \\sqrt{-1}$. Therefore, the zeros of the function are $\textbf{±}\\sqrt{5}$ and $\textbf{±}i$. The function's zeros represent the x-values where the function's value is zero, providing critical information about its behavior, such as where it crosses the x-axis or touches the x-axis. It is the most important point of the function.

Understanding Complex Numbers in the Zeros

Now, let's talk a bit about those complex numbers, $i$. When we encountered $x^2 = -1$, we knew we were dealing with non-real solutions. These are called complex numbers. Complex numbers are numbers that can be expressed in the form $a + bi$, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit ($\sqrt{-1}$). In our case, the solutions $\textbf{±}i$ can be written as $0 + 1i$ and $0 - 1i$. These complex roots always come in conjugate pairs when the coefficients of the polynomial are real numbers, which is the case in our function. This means if $a + bi$ is a root, then $a - bi$ is also a root. Complex roots are a critical part of understanding polynomial functions, especially for higher-degree equations like the one we started with. They help us paint a complete picture of the function's behavior, even if the graph doesn't cross the x-axis at these points. Complex roots provide a more complete understanding of where the function is equal to zero. They influence the overall shape of the function and its behavior across the complex plane. These kinds of root tell us about the function's behavior, especially its relationship to the x-axis. Complex numbers expand the concept of roots beyond just the points where the graph intersects the x-axis; instead, they help us understand the function across a broader domain. So, understanding complex numbers is essential in this context. It allows us to fully grasp the solutions. The appearance of complex numbers also highlights that our function's graph doesn't intersect the x-axis at the complex roots, but these roots are still essential for a complete solution. Complex numbers may seem abstract, but they are vital to the mathematical framework. They are integral in fields like signal processing, electrical engineering, and quantum mechanics. So, getting familiar with them is a good move!

The Answer and its Explanation

So, what's the correct answer, guys? The zeros of the function $f(x) = x^4 - 4x^2 - 5$ are $\textbf{±}\\sqrt{5}$ and $\textbf{±}i$. That means the correct answer choice is A. $\textbf{±}\\sqrt{5}$, $\textbf{±}i$. Let's quickly recap how we got there. We started with a quartic function, made a substitution to turn it into a quadratic equation, solved the quadratic equation for $y$, and then substituted back to find the values of $x$. The values of $x$ where the function equals zero are exactly the zeros of the function. These points are important in understanding the characteristics of the function and how it behaves. The roots $\textbf{±}\\sqrt{5}$ are real, and they represent the points where the graph of the function crosses the x-axis. The roots $\textbf{±}i$ are complex, and they don't appear as x-intercepts on the graph. However, they are still solutions to the equation and contribute to the function's overall behavior. They provide a complete solution that encompasses all possible roots. Getting the right answer is just the first step. The real value comes from understanding the process and why we made the choices we did. Looking at the other answer choices, we can see why they are incorrect. Options B, C, and D are incorrect because they don't match the solutions we found. They don't satisfy the original equation when substituted into the function. This process teaches us about problem-solving, but also gives us a stronger understanding of functions and complex numbers. Remember, when you're tackling problems like this, always double-check your work. Ensure you've solved for the correct variable and that your solutions satisfy the original equation. If the equation looks a bit tricky, always try to simplify it. Making mistakes is ok. That is how we improve!

Practical Applications and Further Learning

Finding the zeros of a function isn't just an exercise in algebra; it's a fundamental skill with lots of real-world applications. For example, in physics, you might use this to calculate the points where a projectile's height is zero (i.e., when it hits the ground). In engineering, these concepts are essential for designing circuits and analyzing signals. Beyond just finding zeros, understanding functions in this way lays the groundwork for more advanced mathematical concepts, such as calculus. You'll often encounter these techniques in calculus when analyzing derivatives and integrals. The process of simplifying equations, using substitutions, and solving for unknowns is a key component of many areas of mathematics. For further study, consider exploring topics like polynomial factorization, the quadratic formula, and the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This means that a polynomial of degree n will have n roots, counting multiplicity. The roots can be real or complex. This also explains why functions of this form can have up to four roots. There are many free online resources, such as Khan Academy, that can help you dive deeper into these topics. Working through different examples and practice problems will help you sharpen your skills and gain confidence. Don't be afraid to ask for help or consult with a teacher or tutor if you're struggling with a concept. Remember, practice makes perfect, and the more you work with functions, the better you'll understand them! So keep practicing, keep exploring, and have fun with it! The world of math is vast and full of interesting things to discover. Always keep a curious mind and enjoy the journey.