Solving For Roots: 5x^4 - 2x + 10x = 0 Equation

Hey guys! Today, we're diving into the fascinating world of equations, and we're tackling a specific one: 5x^4 - 2x + 10x = 0. Our mission? To find the root of this equation. Now, what exactly is a root? Simply put, it's the value of 'x' that makes the equation true. We've got a few options to choose from: A) x = 0, B) x = 1, C) x = -1, and D) x = 2. Let's break it down step by step and figure out the correct answer together!

Understanding the Equation

Before we jump into plugging in numbers, let's take a closer look at our equation: 5x^4 - 2x + 10x = 0. Notice anything interesting? We've got a term with x raised to the power of 4 (that's the 5x^4), and we also have two 'x' terms (-2x and +10x). The first thing we can do to simplify things is combine those 'x' terms. So, -2x + 10x becomes +8x. This means our equation now looks like this: 5x^4 + 8x = 0. Simplifying equations is like decluttering your room – it makes everything much easier to handle!

Why Simplify?

Simplifying isn't just about making the equation look prettier. It actually makes it easier to solve. By combining like terms, we reduce the number of individual pieces we need to deal with. In this case, combining -2x and +10x into 8x makes the next steps much clearer. We're essentially tidying up the mathematical expression so that we can see the solution more easily. Think of it as organizing your ingredients before you start cooking – it prevents you from getting lost in the process!

The Power of Factoring

Now, we've got 5x^4 + 8x = 0. What's the next move? This is where a cool technique called factoring comes into play. Factoring is like finding the common building blocks within an expression. In our equation, both terms have 'x' in them. We can actually pull an 'x' out, which is like dividing both parts of the equation by 'x'. When we do that, we get: x(5x^3 + 8) = 0. See what we did there? We've rewritten the equation in a way that shows 'x' multiplied by another expression. This is a crucial step because it helps us identify potential roots.

What Does Factoring Tell Us?

Factoring is a powerful tool because it uses a simple but fundamental rule of mathematics: If the product of two things is zero, then at least one of those things must be zero. In our case, we have 'x' multiplied by '(5x^3 + 8)'. If the whole thing equals zero, then either 'x' itself is zero, or the expression '(5x^3 + 8)' is zero. This gives us our first potential root immediately! If x = 0, then the entire equation becomes 0. So, option A (x = 0) is looking pretty good right now. But let's not stop there – we need to be thorough and check the other options too.

Testing the Alternatives

We've already seen that factoring gave us a strong hint that x = 0 might be a root. But to be absolutely sure, and to rule out the other options, we need to test each alternative in the original equation. This is like double-checking your answers on a test – you want to be confident in your solution!

Option A: x = 0

Let's plug x = 0 into our simplified equation, 5x^4 + 8x = 0. This gives us: 5(0)^4 + 8(0) = 0. Anything multiplied by zero is zero, so we have 0 + 0 = 0. This is definitely true! So, x = 0 is indeed a root of the equation. We were right on track with our factoring strategy!

Option B: x = 1

Now, let's try x = 1. Plugging this into our simplified equation gives us: 5(1)^4 + 8(1) = 0. Since 1 raised to any power is still 1, this becomes: 5(1) + 8(1) = 0, which simplifies to 5 + 8 = 0. But 5 + 8 is 13, not 0. So, x = 1 is not a root of our equation. We can confidently cross this one off the list.

Option C: x = -1

Next up, let's test x = -1. Plugging this into our simplified equation gives us: 5(-1)^4 + 8(-1) = 0. A negative number raised to an even power becomes positive, so (-1)^4 is 1. This gives us: 5(1) + 8(-1) = 0, which simplifies to 5 - 8 = 0. But 5 - 8 is -3, not 0. So, x = -1 is also not a root of our equation.

Option D: x = 2

Finally, let's try x = 2. Plugging this into our simplified equation gives us: 5(2)^4 + 8(2) = 0. 2 raised to the power of 4 is 16, so this becomes: 5(16) + 8(2) = 0, which simplifies to 80 + 16 = 0. But 80 + 16 is 96, definitely not 0. So, x = 2 is not a root of our equation either.

The Verdict

We've put each option to the test, and the results are in! Options B, C, and D didn't work out – they didn't make the equation true when we plugged them in. But option A, x = 0, passed with flying colors! So, the root of the equation 5x^4 - 2x + 10x = 0 is indeed x = 0. We solved it, guys!

Why is x = 0 a Root?

The reason x = 0 works is pretty straightforward. If you substitute 0 for x in the equation, every term that contains x becomes 0. This is because any number multiplied by 0 is 0. In our original equation, 5x^4 - 2x + 10x = 0, when x = 0, we get 5(0)^4 - 2(0) + 10(0) = 0, which simplifies to 0 - 0 + 0 = 0. This makes the equation true, confirming that x = 0 is a root.

Other Possible Roots

Now, you might be wondering, could there be other roots? Our equation has a term with x raised to the power of 4 (the 5x^4). This tells us that the equation could potentially have up to four roots. We've found one root (x = 0), but there might be others lurking out there. Finding those other roots might require more advanced techniques, like dealing with complex numbers or using numerical methods. But for this specific question, we were given a set of alternatives, and we found the one that works!

Key Takeaways

So, what have we learned on this mathematical adventure? First, we saw the importance of simplifying equations. Combining like terms made our job much easier. Second, we discovered the power of factoring. It helped us quickly identify a potential root. And third, we learned the value of testing alternatives. Plugging in each option and checking if it makes the equation true is a solid way to confirm our solution. Solving equations is like piecing together a puzzle – each step brings you closer to the final answer. And with a bit of practice and the right tools, you can conquer even the trickiest equations!

The Importance of Checking Your Work

Remember, in math (and in life!), it's always a good idea to double-check your work. Whether it's plugging the solution back into the original equation or using a different method to solve the problem, verifying your answer gives you confidence that you've got it right. Think of it as putting a lock on your treasure chest – you want to make sure your solution is secure!

Practice Makes Perfect

Solving equations is a skill, and like any skill, it gets better with practice. The more you work with equations, the more comfortable you'll become with the different techniques and strategies. Don't be afraid to make mistakes – they're part of the learning process. The important thing is to keep trying, keep exploring, and keep challenging yourself. And who knows, maybe one day you'll be the one explaining equation-solving to others! Keep up the great work, guys!