Solving Quadratic Equation: H^2 - 19h = 0

Hey guys! Today, we're diving into a fun little math problem: solving the quadratic equation h² - 19h = 0. Don't worry, it's not as scary as it looks! We're going to break it down step by step, so you can easily understand how to find the solutions. We'll make sure to express our answers in the simplest form, whether they're integers, proper fractions, or improper fractions. If there’s more than one solution (spoiler alert: there will be!), we’ll separate them with commas. So, grab your pencils, and let’s get started!

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what a quadratic equation is. A quadratic equation is basically an equation that can be written in the general form of ax² + bx + c = 0, where 'a', 'b', and 'c' are constants, and 'x' is our variable. The highest power of the variable in a quadratic equation is always 2. Recognizing this form is the first step in tackling these problems.

In our case, the equation h² - 19h = 0 fits this form perfectly. Here, 'a' is 1 (because we have 1h²), 'b' is -19, and 'c' is 0 (since there's no constant term added or subtracted). Understanding these coefficients helps us choose the right method to solve the equation. There are several ways to solve quadratic equations, but for this particular one, we'll use a method called factoring, which is super efficient when we have a common factor in our terms.

Now, you might be wondering, "Why are we even solving these equations?" Well, quadratic equations pop up in all sorts of real-world scenarios. They can model the trajectory of a ball thrown in the air, calculate areas and dimensions, and even help in financial modeling. So, mastering the art of solving them is a pretty valuable skill! Plus, it's like a puzzle, and who doesn't love a good puzzle?

Method 1: Solving by Factoring

Now, let's get down to business and solve our equation h² - 19h = 0. Factoring is our go-to method here, and it's actually quite straightforward. The main idea behind factoring is to break down the equation into simpler parts that we can easily solve. Think of it like taking apart a LEGO creation – we're going to disassemble the equation into its fundamental pieces.

So, the first thing we need to do is identify the common factor in our equation. Looking at h² - 19h, can you spot something that both terms share? That's right, it's 'h'! Both h² and -19h have 'h' in them. This is our ticket to factoring. We can factor out 'h' from the equation, which means we rewrite the equation as a product of 'h' and another expression. When we factor out 'h', we're essentially dividing each term by 'h' and placing 'h' outside the parentheses.

Here’s how it looks: h(h - 19) = 0. See what we did there? We pulled out an 'h' from both terms. Now, we have 'h' multiplied by the expression '(h - 19)'. This is a crucial step because it sets us up to use the zero-product property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In simpler terms, if we have two things multiplied together that equal zero, then either the first thing is zero, the second thing is zero, or both are zero. This is a powerful rule that helps us solve for 'h'.

Applying the Zero-Product Property

Okay, so we've successfully factored our equation into h(h - 19) = 0. Now comes the really cool part – applying the zero-product property. Remember, this property tells us that if the product of two factors is zero, then at least one of the factors has to be zero. In our case, the two factors are 'h' and '(h - 19)'. So, this means that either h = 0 or (h - 19) = 0, or possibly both!

Let's break this down into two separate mini-equations and solve each one individually. First, we have h = 0. Well, that was easy! We've already found one solution: h = 0. This is a straightforward solution, and it’s an integer, just like the question asked for. Sometimes, the solutions are right in front of us, and this is one of those times. Pat yourself on the back for spotting that one!

Now, let’s tackle the second part: (h - 19) = 0. To solve for 'h' in this equation, we need to isolate 'h' on one side of the equation. This means we want to get 'h' all by itself. The way we do that is by adding 19 to both sides of the equation. This is because adding 19 to (h - 19) will cancel out the -19, leaving us with just 'h'. Remember, whatever we do to one side of the equation, we have to do to the other side to keep things balanced. It's like a mathematical see-saw – we need to keep both sides level.

So, adding 19 to both sides gives us: h - 19 + 19 = 0 + 19, which simplifies to h = 19. Voila! We've found our second solution. And guess what? It’s also an integer. So, we have two solutions to our quadratic equation, and they're both nice, whole numbers. High five!

Expressing the Solutions

We've done the hard work of solving the equation, but now we need to make sure we express our solutions in the correct format. The question asked us to write each solution as an integer, proper fraction, or improper fraction in simplest form. And if there are multiple solutions, we need to separate them with commas. Luckily for us, both of our solutions are integers, which are already in their simplest form. We don't need to worry about any fractions this time.

Our solutions are h = 0 and h = 19. To express them as requested, we simply list them separated by a comma. So, the final answer is: 0, 19. That’s it! We’ve successfully solved the quadratic equation and presented our solutions in the correct format. Give yourself a pat on the back – you’ve earned it!

Checking Our Work

Now, before we declare victory and move on to the next math adventure, let’s do a quick check to make sure our solutions are correct. This is always a good practice, no matter how confident we feel. Checking our work helps us catch any silly mistakes and ensures that we’re on the right track. It’s like proofreading a paper before submitting it – a little extra effort can go a long way.

To check our solutions, we’ll plug each one back into the original equation, h² - 19h = 0, and see if it holds true. If both sides of the equation are equal after we substitute 'h', then we know our solution is correct. Let’s start with the first solution, h = 0. Substituting this into the equation, we get: 0² - 19(0) = 0. This simplifies to 0 - 0 = 0, which is indeed true. So, h = 0 is definitely a solution. Nice!

Now, let’s check the second solution, h = 19. Substituting this into the equation, we get: 19² - 19(19) = 0. Now, 19² is 361, so we have 361 - 19(19) = 0. And 19 multiplied by 19 is also 361, so we have 361 - 361 = 0, which simplifies to 0 = 0. This is also true! So, h = 19 is another valid solution. Woohoo!

Since both of our solutions check out, we can be confident that we’ve solved the equation correctly. This extra step of checking our work gives us peace of mind and reinforces our understanding of the problem. It’s like putting the final piece in a jigsaw puzzle – everything clicks into place, and we can see the complete picture.

Conclusion

Alright, guys! We’ve successfully solved the quadratic equation h² - 19h = 0. We used the method of factoring to break down the equation, applied the zero-product property to find our solutions, and expressed them in the correct format. We even double-checked our work to make sure everything was spot on. You've tackled this problem like a champ!

The solutions we found were h = 0 and h = 19. These are the values of 'h' that make the equation true. Remember, solving quadratic equations is a valuable skill that can be applied in various real-world scenarios. Whether you're calculating the trajectory of a projectile or modeling financial growth, understanding how to solve these equations will come in handy.

So, keep practicing, keep exploring, and keep those math muscles strong. Quadratic equations might seem intimidating at first, but with a little practice and a step-by-step approach, you can conquer them all. And who knows, maybe you'll even start to enjoy them! Math is like a puzzle, and every solved equation is a victory. Keep up the great work, and I'll catch you in the next math adventure!