Solving Quartic Equations: Factoring $x^4 - 5x^2 - 36 = 0$

Hey guys! Let's dive into solving the equation $x^4 - 5x^2 - 36 = 0$. This is a quartic equation, which means it has a variable raised to the fourth power. Don't freak out, because we can solve this using a technique called factoring. Factoring is like the cool shortcut to finding the solutions (also known as roots or zeros) of an equation. The goal is to rewrite the equation as a product of simpler expressions. Once we've done that, we can easily find the values of x that make the original equation true.

In this case, we're going to treat this equation as if it were a quadratic equation, but with a little twist. You'll see how factoring comes in handy, and we'll walk through each step so you can master this type of problem. Keep in mind that understanding factoring is super important because it is a fundamental concept in algebra and is used in many other areas of mathematics. So, buckle up, and let's get started! We will learn how to break down this equation into manageable pieces, which makes finding the solutions much easier. The main idea is to transform the given quartic equation into something that resembles a quadratic equation, allowing us to apply the factoring method we're familiar with. Factoring is basically the reverse of expanding, where we take an expression and break it down into its component parts. The cool thing is that it allows us to find the values of the variable(s) that make the equation true without having to resort to more complicated methods like the quadratic formula (although that would work too!).

So, let's go through this equation step by step, making sure we fully understand each element. I'll make sure to break down the process into simple terms, removing all the confusing jargon, so you can follow along. If you feel like you are a bit rusty in factoring, don't worry, we will go over some basic concepts to get you back on track. The key thing to remember is to practice. The more examples you go through, the better you'll become. It's like riding a bike; the more you do it, the easier it gets. The most important thing is that you're trying. Don't be afraid to make mistakes because that's how we learn. Keep in mind that the idea behind factoring this kind of equation is to turn it into a product of factors, so that we can easily find the values of x that make the equation true. In this equation, the key is to realize that we can treat x^2 as a single variable because the equation only involves x^2 and x^4. By doing this, the equation becomes easier to understand. The main idea is to identify how to rewrite our original expression as a product of simpler expressions. This step is critical in helping us solve the equation efficiently. Factoring is a basic yet effective technique.

Step-by-Step Factoring Process

Alright, let's jump right into solving the equation $x^4 - 5x^2 - 36 = 0$. The secret to tackling this is to recognize that $x^4$ is the same as $(x^2)^2$. This allows us to treat the equation like a quadratic equation.

1. Substitution: Let's make our lives a bit easier. We'll substitute $y = x^2$. This changes our equation to $y^2 - 5y - 36 = 0$. See? It looks like a standard quadratic equation now.

2. Factoring the Quadratic: Now, we need to factor the quadratic equation $y^2 - 5y - 36 = 0$. We're looking for two numbers that multiply to -36 and add up to -5. Those numbers are -9 and 4. So, we can factor the equation as $(y - 9)(y + 4) = 0$. This is where we can use the basic rules of factoring to break down a quadratic expression into its constituent parts. The objective is to convert the given expression into a product of two binomials that are equal to zero. Remember that the factored form is essentially the opposite of what you would get when you expand the product of two binomials. Therefore, you're doing the reverse of a process you likely already know. If you find it difficult to find the two numbers, there are a few techniques that might help, such as listing the factors of the constant term (-36) and seeing which pair adds up to the coefficient of the linear term (-5). Also, note that factoring can sometimes be tricky, but with practice, you'll get better at recognizing patterns and identifying the factors. This is the core of the factoring technique, transforming our equation into a form where we can easily find the roots. Practice is the best way to master this skill!

3. Solve for y: From the factored form $(y - 9)(y + 4) = 0$, we can find the values of y that make the equation true. We set each factor equal to zero:

   • $y - 9 = 0$ which gives us $y = 9$.
   • $y + 4 = 0$ which gives us $y = -4$. Here, we get the values of y (which is $x^2$) that satisfy the equation. We have $y = 9$ and $y = -4$. Remember, our initial substitution was y = x². This means we're not done yet; we need to find the values of x. We're getting closer to our final answers! Keep in mind that each solution of y will give us two possible values for x (except in specific scenarios). We can see that finding the roots involves isolating the variable to get the values that make the whole equation true. This process is much simpler and more manageable than trying to solve the original quartic equation directly.

4. Solve for x: Now, we need to go back to our original variable, x. We know that $y = x^2$.

   • If $y = 9$, then $x^2 = 9$. Taking the square root of both sides, we get x = oxed{\pm 3}.
   • If $y = -4$, then $x^2 = -4$. Taking the square root of both sides, we get $x = \pm \sqrt{-4} = \pm 2i$, where i is the imaginary unit ($\sqrt{-1}$).

So, the solutions to the equation are $x = 3$, $x = -3$, $x = 2i$, and $x = -2i$. This is where we find the actual values of x that make the original equation true. Recall that the square root of a negative number introduces complex numbers, which include a real and imaginary part. The imaginary unit i is a crucial concept in solving this equation since it allows us to account for the negative values inside the square root. By going back to our original variable, x, we can now provide a complete answer for the equation. Remember, the key is to understand each step and the reason behind it. This is where all the pieces of the puzzle fit together to give us the solutions for the equation.

The Answer and Why It Matters

The correct answer is C. $x = \pm 2$ and $x = \pm 3i$. This is because we found that the solutions to the equation $x^4 - 5x^2 - 36 = 0$ are $x = 3$, $x = -3$, $x = 2i$, and $x = -2i$. Understanding how to solve quartic equations by factoring is important for several reasons. It strengthens your algebra skills, which is fundamental for advanced math and science courses. It also introduces you to complex numbers, an important concept in many areas of mathematics and engineering.

In summary, solving this quartic equation using factoring involves recognizing the quadratic structure, performing a substitution to simplify the equation, factoring the quadratic, and finally solving for the original variable. The process reinforces the importance of breaking down a problem into simpler parts and understanding the underlying concepts. Always remember to double-check your work at each step to minimize errors and to gain a better grasp of the procedure. Practice more examples. This will help you understand and solve other more complex equations in the future.

Hope this was helpful, and keep up the awesome work! Remember to keep practicing and exploring different types of equations.