Solving Quadratic Equations: Finding The Values Of X

Hey guys! Let's dive into the world of quadratic equations! We're going to break down how to solve them and find those elusive values of x that make the equation true. It's not as scary as it sounds, I promise! We'll start with the given equation: $3x^2 - 27x = 0$. Our mission? To find the values of x that satisfy this equation. Let's get started, shall we?

Understanding Quadratic Equations

Alright, before we jump into the nitty-gritty, let's chat about what a quadratic equation even is. In its most basic form, a quadratic equation looks like this: $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to zero. See that x raised to the power of 2? That's the key characteristic of a quadratic equation. It tells us we're likely to have two solutions (or roots) for x. These solutions are the values that, when plugged back into the equation, make it all balance out to zero. The cool thing about quadratics is they pop up everywhere in the real world. Think about the trajectory of a ball thrown in the air, the shape of a bridge, or even the path of a bouncing basketball. They are super important for a lot of reasons. Solving quadratic equations is a fundamental skill in algebra and opens the door to more advanced math concepts. This is why this question is so important, because it allows us to show our critical thinking skills.

Now, back to our equation: $3x^2 - 27x = 0$. Notice how it fits the general form? We have an x squared term, an x term, and a constant term (which, in this case, is zero). Our goal is to manipulate the equation to isolate x and find its possible values. There are several ways to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. For this particular equation, factoring is the most straightforward approach. Because factoring is the approach we're going to take here, make sure that we pay close attention to the concepts, so we can nail it.

Let's get cracking with how to solve this equation! Factoring is like detective work, but instead of finding a criminal, you're finding factors that multiply together to give you the original equation. It's all about breaking down the equation into smaller, more manageable parts. We need to find what values of x when we replace them in the equation will make the equation true.

Factoring the Equation

So, let's take our equation: $3x^2 - 27x = 0$. The first step in factoring is to look for the greatest common factor (GCF). The GCF is the largest number or expression that divides evenly into all terms of the equation. In our equation, the GCF is 3x. Notice that both terms, $3x^2$ and $-27x$, are divisible by 3x. Let's factor out the 3x:

$$3x(x - 9) = 0$$

See how we've rewritten the equation? Now, we have a product of two factors, $3x$ and $(x - 9)$, that equals zero. This is super useful because of the zero-product property. 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. This is the crucial step in solving for x. It gives us a way to break down the equation into simpler parts. We can apply this property to our factored equation.

Now, let's apply the zero-product property. We have two factors: $3x$ and $(x - 9)$. This means either $3x = 0$ or $(x - 9) = 0$.

Let's solve each of these equations separately.

Solving for x

Alright, let's tackle our first equation: $3x = 0$. To solve for x, we need to isolate it. We can do this by dividing both sides of the equation by 3:

$$\frac{3x}{3} = \frac{0}{3}$$

$$x = 0$$

So, one solution for x is 0. Great! Now, let's move on to the second equation: $(x - 9) = 0$. To isolate x, we need to add 9 to both sides of the equation:

$$x - 9 + 9 = 0 + 9$$

$$x = 9$$

And there you have it! Our second solution for x is 9. Therefore, the values of x that make the equation $3x^2 - 27x = 0$ true are $x = 0$ and $x = 9$. We have successfully factored the quadratic equation, applied the zero-product property, and solved for x. Yay, us!

Checking Your Answers

It's always a good idea to double-check your answers, just to be absolutely sure. We can do this by plugging each of our solutions back into the original equation and see if it holds true. Let's start with $x = 0$:

$$3(0)^2 - 27(0) = 0$$

$$0 - 0 = 0$$

$$0 = 0$$

Looks good! Now, let's check $x = 9$:

$$3(9)^2 - 27(9) = 0$$

$$3(81) - 243 = 0$$

$$243 - 243 = 0$$

$$0 = 0$$

Excellent! Both solutions satisfy the original equation. We've done it! We have successfully solved the quadratic equation and found the correct values for x. This ability is super important when we look at the trajectory of a ball, or the structure of a bridge. This process is used across all sorts of disciplines.

Choosing the Correct Answer

Now that we've solved the equation, let's look at the multiple-choice options and see which one matches our solutions. Our solutions are $x = 0$ and $x = 9$.

Looking at the options:

A) $x = 0$ or $x = 3$ - Nope, we got 9, not 3. B) $x = 0$ or $x = 9$ - Ding, ding, ding! This is our answer. C) $x = 3$ or $x = 9$ - Nope, we have 0, not 3. D) $x = 3$ or - Nope, we have 0, not 3.

So, the correct answer is B) $x = 0$ or $x = 9$. Congrats! You've successfully solved a quadratic equation and selected the right answer.

Conclusion

Solving quadratic equations might seem tricky at first, but with practice, it becomes much easier. Remember the key steps: factor, apply the zero-product property, and solve for x. Always double-check your answers. Keep practicing, and you'll become a pro in no time! Keep in mind that math can be fun! Also, there are many resources out there to assist you. Always make sure to check your answers too.

Great job everyone! You've successfully tackled a quadratic equation. Keep up the excellent work, and always keep learning. This should help you on your future tests, and in your day-to-day life. Math is all around us, and this process will help you break down complex concepts into easier ones. Keep up the great work, everyone!