Solving 3x² - 12x + 24 = 0 A Step-by-Step Guide

Hey everyone! Today, we're diving into the fascinating world of quadratic equations, specifically tackling the equation 3x² - 12x + 24 = 0. Now, quadratic equations might sound intimidating, but trust me, they're super manageable once you understand the basic principles. This article will guide you through the process of solving this equation, step by step, and we'll explore the different methods you can use. We'll also touch on the underlying concepts and why these methods work. So, whether you're a student grappling with algebra or just someone curious about math, this is the perfect place to start. Let's get those brain cells firing!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's quickly recap what a quadratic equation actually is. In its most general form, a quadratic equation looks like this: ax² + bx + c = 0, where 'a', 'b', and 'c' are constants (numbers) and 'x' is the variable we're trying to find. The highest power of 'x' in a quadratic equation is always 2, which is why it's called a quadratic equation. These equations pop up everywhere in math and science, from calculating trajectories to modeling curves.

Think of 'a' as the coefficient of the x² term, 'b' as the coefficient of the x term, and 'c' as the constant term. In our equation, 3x² - 12x + 24 = 0, we have a = 3, b = -12, and c = 24. Notice the negative sign in front of the 12; it's crucial to keep track of these signs! The solutions to a quadratic equation, also known as the roots or zeros, are the values of 'x' that make the equation true. In other words, they're the points where the graph of the quadratic function (a parabola) intersects the x-axis. Sometimes, a quadratic equation will have two real solutions, sometimes one, and sometimes none (in the realm of real numbers). We'll see how this plays out as we solve our example.

Now, why are quadratic equations so important? Well, they're everywhere! They model the path of a ball thrown in the air, the shape of satellite dishes, and even the design of bridges. Understanding how to solve them opens up a whole world of applications. And the best part? There are several different methods you can use, each with its own strengths and weaknesses. We'll focus on two primary methods for solving our equation: factoring (if possible) and the quadratic formula. So, buckle up and let's dive in!

Method 1: Simplifying the Equation

The very first thing we should always do when faced with any equation, not just quadratic ones, is to look for opportunities to simplify it. This can make the subsequent steps much easier and reduce the chances of making errors. In our case, we have the equation 3x² - 12x + 24 = 0. Take a close look at the coefficients: 3, -12, and 24. Do you notice anything they have in common? That's right, they're all divisible by 3! This means we can simplify the equation by dividing both sides by 3. Remember, whatever we do to one side of an equation, we must do to the other to maintain the equality.

Dividing each term by 3, we get: (3x²)/3 - (12x)/3 + 24/3 = 0/3. This simplifies to x² - 4x + 8 = 0. Wow, that looks much cleaner, doesn't it? We've effectively reduced the coefficients, making the equation easier to work with. This step is crucial because it often reveals simpler patterns or allows us to use factoring techniques more easily. It's like decluttering your workspace before tackling a big project; you'll be surprised how much clearer things become! Now, we have a simplified quadratic equation, x² - 4x + 8 = 0, that we can solve using various methods. We've already made significant progress by simplifying, and this will pave the way for a smoother solving process. Always remember to look for opportunities to simplify equations – it's a fundamental skill in algebra and beyond.

Method 2: Applying the Quadratic Formula

When factoring doesn't quite cut it, or when you're faced with a quadratic equation that seems stubbornly resistant to simple factorization, the quadratic formula is your trusty sidekick. It's a universal solution that works for any quadratic equation in the form ax² + bx + c = 0. The quadratic formula is given by:

x = (-b ± √(b² - 4ac)) / 2a

Don't let the formula intimidate you! It might look a bit complex at first, but it's just a matter of plugging in the correct values. Remember, we've already simplified our equation to x² - 4x + 8 = 0, where a = 1, b = -4, and c = 8. Now, let's carefully substitute these values into the quadratic formula.

Substituting, we get:

x = (-(-4) ± √((-4)² - 4 * 1 * 8)) / (2 * 1)

Let's break this down step by step. First, -(-4) becomes 4. Then, (-4)² is 16. Inside the square root, we have 4 * 1 * 8, which is 32. So, our equation now looks like:

x = (4 ± √(16 - 32)) / 2

Next, we simplify inside the square root: 16 - 32 = -16. This is where things get interesting! We now have the square root of a negative number: √(-16). In the realm of real numbers, the square root of a negative number is undefined. However, we can venture into the world of complex numbers to handle this. The square root of -1 is denoted as 'i' (the imaginary unit). So, √(-16) can be written as √(16 * -1) = √(16) * √(-1) = 4i.

Our equation now becomes:

x = (4 ± 4i) / 2

Finally, we can simplify by dividing both terms in the numerator by 2:

x = 2 ± 2i

So, the solutions to our quadratic equation are x = 2 + 2i and x = 2 - 2i. These are complex solutions, meaning they have both a real part (2) and an imaginary part (2i and -2i). This tells us that the graph of the quadratic function x² - 4x + 8 = 0 does not intersect the x-axis in the real number plane. It's a beautiful example of how the quadratic formula can reveal the nature of the solutions, even when they're not real numbers. Guys, the quadratic formula might seem like a beast, but once you tame it, it's an incredibly powerful tool!

Nature of the Roots: The Discriminant

As we've seen, quadratic equations can have different types of solutions: two real solutions, one real solution (a repeated root), or two complex solutions. But how can we know what kind of solutions to expect before we actually solve the equation? This is where the discriminant comes in handy. The discriminant is a part of the quadratic formula that sits under the square root sign: b² - 4ac. The value of the discriminant tells us a lot about the nature of the roots.

• If b² - 4ac > 0, the equation has two distinct real solutions.
• If b² - 4ac = 0, the equation has one real solution (a repeated root).
• If b² - 4ac < 0, the equation has two complex solutions (as we saw in our example).

Let's calculate the discriminant for our equation, x² - 4x + 8 = 0, where a = 1, b = -4, and c = 8. The discriminant is: (-4)² - 4 * 1 * 8 = 16 - 32 = -16. Since the discriminant is negative (-16), we know that the equation has two complex solutions, which confirms what we found using the quadratic formula.

Understanding the discriminant is a powerful shortcut. It allows you to quickly determine the type of solutions you'll encounter, saving you time and effort. It's like having a weather forecast for your math problems – you know what kind of solution weather to expect!

Visualizing the Solutions

It's often helpful to visualize what's happening with quadratic equations graphically. The graph of a quadratic equation ax² + bx + c = 0 is a parabola. The solutions to the equation are the x-intercepts of the parabola – the points where the parabola crosses the x-axis.

• If the parabola intersects the x-axis at two points, the equation has two real solutions.
• If the parabola touches the x-axis at one point (the vertex), the equation has one real solution (a repeated root).
• If the parabola doesn't intersect the x-axis at all, the equation has two complex solutions.

In our case, the solutions are complex, meaning the parabola doesn't intersect the x-axis. This makes sense because we found that the discriminant was negative. Imagine a parabola sitting above the x-axis, never quite touching it – that's the visual representation of our equation's solutions. Visualizing quadratic equations helps to solidify your understanding and connect the algebraic solutions with their graphical counterparts. It's like seeing the music instead of just hearing the notes!

Conclusion

So, guys, we've journeyed through the process of solving the quadratic equation 3x² - 12x + 24 = 0. We started by simplifying the equation, then we fearlessly applied the quadratic formula to find the solutions: x = 2 + 2i and x = 2 - 2i. We also explored the concept of the discriminant and how it helps us predict the nature of the roots. And finally, we visualized the solutions graphically, connecting the algebra to the geometry.

Quadratic equations might have seemed daunting at first, but hopefully, this guide has demystified them. Remember, practice makes perfect! The more you solve these equations, the more comfortable you'll become with the different methods and concepts. Keep exploring, keep learning, and keep those math muscles strong! Understanding quadratic equations is a fundamental skill in mathematics and opens doors to a wide range of applications in science, engineering, and beyond. So, keep up the great work, and you'll be conquering quadratic equations in no time!