Solving The Equation (11u)(3u - 5) = 0: A Step-by-Step Guide

Hey guys! Let's dive into solving this equation: $(11u)(3u - 5) = 0$. This might look a little intimidating at first, but trust me, it's totally manageable. We're going to break it down step by step, so you'll be a pro at solving these kinds of problems in no time. Our main keywords here are solving equations, specifically this particular quadratic form. Remember, the key to mastering math is understanding the fundamentals, so let's get started!

Understanding the Zero Product Property

Before we jump into the nitty-gritty, let's talk about a super important concept called the Zero Product Property. This property is the backbone of how we'll solve this equation. Simply put, the Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. Make sense? Let's write it out in a more mathematical way:

If $A * B = 0$, then either $A = 0$ or $B = 0$ (or both!).

This might seem like a no-brainer, but it's incredibly powerful. Think about it – if you multiply two numbers and get zero, one of them has to be zero. There's no other way! This principle allows us to take complex equations and break them down into simpler ones. In our case, we have $(11u)(3u - 5) = 0$. We can think of $(11u)$ as our 'A' and $(3u - 5)$ as our 'B'. So, according to the Zero Product Property, either $(11u)$ must equal zero, or $(3u - 5)$ must equal zero. This is a huge step because now we can create two separate, easier equations to solve. Remember this Zero Product Property; it's your best friend when tackling these problems.

Now, why is this so important? Well, it turns a seemingly complicated problem into two simple ones. We go from dealing with a product of two expressions to dealing with each expression individually. This makes our lives way easier. Plus, understanding this property unlocks the door to solving a whole bunch of other equations, too. It's a fundamental concept in algebra, so getting a solid grasp on it now will pay off big time later on. We will use Zero Product Property to find each value of the variable u.

Breaking Down the Equation: Applying the Zero Product Property

Okay, now that we've got the Zero Product Property under our belts, let's apply it to our equation: $(11u)(3u - 5) = 0$. Remember, we identified $(11u)$ as 'A' and $(3u - 5)$ as 'B'. So, according to the property, we can set each of these equal to zero:

1. $11u = 0$
2. $3u - 5 = 0$

See how we've transformed one equation into two? This is the magic of the Zero Product Property in action! Now, we have two much simpler equations to solve individually. Let's start with the first one: $11u = 0$. To isolate 'u', we need to get rid of that 11. Since it's being multiplied by 'u', we'll do the opposite operation: division. We'll divide both sides of the equation by 11:

$11u / 11 = 0 / 11$

This simplifies to:

$u = 0$

Boom! We've found our first solution. Now, let's move on to the second equation: $3u - 5 = 0$. This one's a little trickier, but don't worry, we've got this. Our goal is still the same: to isolate 'u'. First, we need to get rid of the -5. To do that, we'll add 5 to both sides of the equation:

$3u - 5 + 5 = 0 + 5$

This simplifies to:

$3u = 5$

Now, we have $3u = 5$. Just like before, we need to get 'u' by itself. Since 3 is being multiplied by 'u', we'll divide both sides by 3:

$3u / 3 = 5 / 3$

This gives us:

$u = 5/3$

And there you have it! We've found our second solution. Remember to always double-check your work, especially when dealing with fractions. These individual equations are much simpler to tackle, and breaking down the original equation using the Zero Product Property is key to solving it. Practice this technique, and you'll be solving equations like a math whiz!

Solving for u: Step-by-Step Solutions

Alright, let's recap and formally solve each of the equations we set up in the previous section. We're focusing on solving for u, which is our variable. Remember, our two equations were:

1. $11u = 0$
2. $3u - 5 = 0$

Let's tackle the first one: $11u = 0$. As we discussed, to isolate 'u', we need to divide both sides of the equation by 11. This is because 11 is the coefficient of 'u', and we want to undo the multiplication. So, we have:

$11u / 11 = 0 / 11$

When we perform the division, we get:

$u = 0$

So, one solution is $u = 0$. This is a pretty straightforward one, right? Any number multiplied by zero is zero, so this makes perfect sense. Now, let's move on to the second equation: $3u - 5 = 0$. This one requires a couple of steps, but don't worry, we'll take it slow and steady.

First, we need to isolate the term with 'u', which is $3u$. To do this, we need to get rid of the -5. The opposite of subtraction is addition, so we'll add 5 to both sides of the equation:

$3u - 5 + 5 = 0 + 5$

This simplifies to:

$3u = 5$

Great! We've got the term with 'u' by itself. Now, we need to get 'u' completely alone. Just like in the first equation, we have a coefficient (3) multiplying 'u'. To undo the multiplication, we'll divide both sides by 3:

$3u / 3 = 5 / 3$

This gives us:

$u = 5/3$

And there's our second solution! So, the solutions to the equation $3u - 5 = 0$ is $u = 5/3$. Remember, it's perfectly okay to have a fractional solution. In fact, they're quite common in algebra. Now we know, step by step, how to isolate u in both cases.

Final Solutions and Verification

Okay, we've done the hard work! We've broken down the equation, applied the Zero Product Property, and solved for 'u' in each case. Now, let's bring it all together and state our final solutions. We found two solutions for the equation $(11u)(3u - 5) = 0$:

1. $u = 0$
2. $u = 5/3$

These are the values of 'u' that will make the equation true. But, before we pat ourselves on the back, it's always a good idea to verify our solutions. This means plugging each solution back into the original equation to make sure it works. It's like a final check to catch any mistakes we might have made along the way. Let's start with $u = 0$. We'll substitute 0 for 'u' in the original equation:

$(11 * 0)(3 * 0 - 5) = 0$

Simplifying, we get:

$(0)(-5) = 0$

$0 = 0$

Yep, that checks out! So, $u = 0$ is definitely a solution. Now, let's verify the second solution, $u = 5/3$. This one might look a little messier, but don't let it intimidate you. We'll substitute $5/3$ for 'u' in the original equation:

$(11 * 5/3)(3 * 5/3 - 5) = 0$

Let's simplify this step by step. First, we'll multiply $11 * 5/3$:

$(55/3)(3 * 5/3 - 5) = 0$

Next, let's simplify the expression inside the second set of parentheses: $3 * 5/3$. The 3's cancel out, leaving us with:

$(55/3)(5 - 5) = 0$

Now, we have:

$(55/3)(0) = 0$

Anything multiplied by zero is zero, so:

$0 = 0$

Awesome! $u = 5/3$ also checks out. So, we can confidently say that our final solutions are $u = 0$ and $u = 5/3$. Verifying our answers is a crucial step in solving equations, as it ensures accuracy and builds confidence in our work. Remember to always take that extra step!

Conclusion: Mastering Quadratic Equations

Woohoo! We did it! We successfully solved the equation $(11u)(3u - 5) = 0$. We started by understanding the Zero Product Property, then we broke down the equation into simpler parts, solved for 'u' in each part, and finally, we verified our solutions. This is a fantastic example of how to tackle quadratic equations that are already factored. Understanding and applying the Zero Product Property is a fundamental skill in algebra, and you've now got it in your toolbox!

The key takeaways here are:

• Zero Product Property: If the product of two or more factors is zero, then at least one of the factors must be zero.
• Breaking down equations: Use the Zero Product Property to transform a single equation into multiple simpler equations.
• Solving for the variable: Isolate the variable by performing the opposite operations (addition/subtraction, multiplication/division).
• Verification: Always plug your solutions back into the original equation to check your work.

Solving equations like this is a building block for more advanced math topics. The more you practice, the more comfortable you'll become with these techniques. Don't be afraid to make mistakes – they're part of the learning process! Keep practicing, keep asking questions, and you'll be a math master in no time. Great job today, guys!