Solving Y=3x²-12x+16: A Step-by-Step Guide

Hey guys! Today, we're going to dive into solving the quadratic equation y = 3x² - 12x + 16. Quadratic equations might seem intimidating at first, but don't worry, we'll break it down step by step so it's super easy to understand. We'll explore different methods, look at the vertex form, and by the end, you'll be a pro at tackling these types of problems. Let's get started!

Understanding Quadratic Equations

Before we jump into solving, let's quickly recap what a quadratic equation actually is. A quadratic equation is a polynomial equation of the second degree. That just means the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. In our specific equation, y = 3x² - 12x + 16, we can see that a = 3, b = -12, and c = 16. Understanding this basic form is crucial because it helps us identify the different parts we'll be working with. Quadratic equations pop up everywhere in math and real-world applications, from physics problems involving projectile motion to engineering designs. So, grasping how to solve them is a seriously valuable skill. We're not just doing abstract math here; we're learning something that can be applied in many different scenarios. Think about designing bridges, predicting the path of a ball thrown in the air, or even optimizing business processes – quadratic equations play a key role in all of these. Therefore, mastering the techniques to solve them will open doors to various fields and problem-solving opportunities. The more comfortable you become with these equations, the more you'll start noticing them in the world around you, which is pretty cool!

Method 1: Completing the Square

One of the most powerful methods for solving quadratic equations is completing the square. This technique allows us to rewrite the equation in vertex form, which gives us a lot of useful information about the parabola (the U-shaped curve that quadratic equations represent when graphed). Okay, so let’s walk through it. The goal of completing the square is to transform the quadratic equation into the form a(x - h)² + k, where (h, k) is the vertex of the parabola. Let's start with our equation: y = 3x² - 12x + 16. First, we factor out the coefficient of the x² term (which is 3) from the first two terms: y = 3(x² - 4x) + 16. Now, we need to complete the square inside the parentheses. To do this, we take half of the coefficient of the x term (-4), square it, and add it inside the parentheses. Half of -4 is -2, and (-2)² is 4. But here's the catch: because we're inside the parentheses, which are being multiplied by 3, we're actually adding 3 * 4 = 12 to the equation. So, we need to subtract 12 outside the parentheses to keep the equation balanced: y = 3(x² - 4x + 4) + 16 - 12. Now we can rewrite the expression inside the parentheses as a perfect square: y = 3(x - 2)² + 4. And boom! We’ve completed the square. The equation is now in vertex form, and we can easily see that the vertex of the parabola is (2, 4). This method not only helps in solving for the roots (if any) but also gives a clear picture of the parabola's key features. Completing the square might seem a little tricky at first, but with practice, you'll find it's an incredibly versatile tool. It’s a method that provides deep insight into the structure of quadratic equations, allowing us to understand their behavior and properties more fully.

Method 2: Using the Quadratic Formula

The quadratic formula is another fantastic tool in our arsenal for solving quadratic equations. This formula works for any quadratic equation, no matter how messy it looks. It's like a universal key that unlocks the solutions! The quadratic formula is derived from the process of completing the square, but it provides a direct way to find the solutions (also known as roots or x-intercepts) without going through all the steps each time. So, what is this magical formula? For a quadratic equation in the form ax² + bx + c = 0, the quadratic formula is: x = [-b ± √(b² - 4ac)] / (2a). Remember, a, b, and c are the coefficients from our quadratic equation. Let's apply this to our equation, y = 3x² - 12x + 16. We identified earlier that a = 3, b = -12, and c = 16. Now we just plug these values into the formula: x = [12 ± √((-12)² - 4 * 3 * 16)] / (2 * 3). Let’s simplify this step by step. First, calculate the discriminant (the part under the square root): (-12)² - 4 * 3 * 16 = 144 - 192 = -48. Aha! We've hit a snag. The discriminant is negative. This means that the quadratic equation has no real roots. The solutions are complex numbers, which involve the imaginary unit 'i' (where i = √-1). So, while the quadratic formula always gives us the solutions, in this case, they’re not real numbers. This is a crucial piece of information! It tells us that the parabola doesn’t intersect the x-axis. The quadratic formula is a fundamental tool in algebra, and understanding how to use it, including interpreting the discriminant, is essential. It provides a reliable method for solving quadratic equations, regardless of their complexity, and it’s a skill that will serve you well in more advanced math courses.

Analyzing the Discriminant

Speaking of the discriminant, let's dive a bit deeper into what it tells us. The discriminant, represented by b² - 4ac (the part under the square root in the quadratic formula), is like a detective that gives us clues about the nature of the roots of a quadratic equation. It tells us whether the equation has two distinct real roots, one repeated real root, or no real roots (complex roots). As we saw in the previous section, the sign of the discriminant is the key. If the discriminant is positive (b² - 4ac > 0), the quadratic equation has two distinct real roots. This means the parabola intersects the x-axis at two different points. Think of it like this: the square root of a positive number gives us two real values (one positive and one negative), leading to two different solutions for x. On the other hand, if the discriminant is zero (b² - 4ac = 0), the quadratic equation has one repeated real root. This means the parabola touches the x-axis at exactly one point – the vertex of the parabola lies on the x-axis. In this case, the ± part of the quadratic formula becomes irrelevant since we’re adding or subtracting zero. Finally, if the discriminant is negative (b² - 4ac < 0), the quadratic equation has no real roots. This is what we encountered with our example equation. It means the parabola does not intersect the x-axis at all. The solutions are complex numbers, involving the imaginary unit 'i'. Understanding the discriminant is incredibly useful because it allows us to quickly determine the type of solutions we'll get without having to go through the entire quadratic formula calculation. It’s a quick check that can save us time and effort and provides valuable insight into the behavior of the quadratic equation. So, always remember to take a peek at the discriminant – it’s a super helpful tool in your mathematical toolkit!

The Vertex Form and Graphing

We touched on the vertex form earlier, but let's really explore why it's so awesome, especially when it comes to graphing quadratic equations. The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola. Remember, the vertex is the turning point of the parabola – it’s either the minimum (if the parabola opens upwards) or the maximum (if the parabola opens downwards). In our case, we completed the square and got the equation in vertex form: y = 3(x - 2)² + 4. This tells us immediately that the vertex is at the point (2, 4). The 'a' value (which is 3 in our equation) tells us about the shape and direction of the parabola. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, it opens downwards. The larger the absolute value of 'a', the narrower the parabola. So, since our 'a' is 3 (positive), our parabola opens upwards and is a bit narrower than the standard parabola (y = x²). Knowing the vertex and the direction the parabola opens is a huge step in graphing the equation. We have a key point (the vertex) and know the general shape. To get an even more accurate graph, we can find a few more points. One way to do this is to plug in some x-values into our equation and see what y-values we get. For example, we can plug in x = 0: y = 3(0 - 2)² + 4 = 3 * 4 + 4 = 16. So, we have the point (0, 16). Similarly, we can plug in x = 4: y = 3(4 - 2)² + 4 = 3 * 4 + 4 = 16. So, we have the point (4, 16). With the vertex and a couple of other points, we can sketch a pretty accurate graph of the parabola. Understanding the vertex form makes graphing quadratic equations much easier. It provides a clear starting point (the vertex) and gives us valuable information about the shape and direction of the parabola. Graphing quadratic equations is not just a visual exercise; it provides a deeper understanding of the equation's behavior and its solutions.

Real-World Applications

Now, let's talk about why all of this matters in the real world. Quadratic equations aren't just abstract mathematical concepts; they show up in all sorts of places! One of the most common applications is in physics, particularly in projectile motion. When you throw a ball, kick a soccer ball, or launch a rocket, the path it follows can be described by a parabola, which is the graph of a quadratic equation. The equation can help us determine the maximum height the object reaches, the distance it travels, and the time it spends in the air. This is super useful for things like designing sports equipment, calculating trajectories for artillery, or even planning space missions! Another area where quadratic equations are used is in engineering. For example, engineers use quadratic equations to design bridges, arches, and other structures. The parabolic shape provides strength and stability, distributing weight evenly. They also use these equations in electrical engineering to analyze circuits and optimize performance. Business and economics also rely on quadratic equations. Companies use them to model costs, revenue, and profit. For example, they might use a quadratic equation to determine the price that will maximize their profit, considering factors like production costs and demand. Optimizing processes, whether in manufacturing, finance, or logistics, often involves solving quadratic equations. Even computer graphics and video games use quadratic equations to create smooth curves and realistic motion. Think about the curves in a character's animation or the trajectory of a projectile in a game – quadratic equations are often behind the scenes making it all happen. Understanding quadratic equations opens up a world of possibilities. It allows us to analyze and solve problems in diverse fields, making it a truly valuable skill to have. So, the next time you see a curved path or think about optimizing something, remember that quadratic equations might be the key!

Conclusion

So, guys, we've covered a lot today! We've explored what quadratic equations are, how to solve them using completing the square and the quadratic formula, how to analyze the discriminant, how to graph them using the vertex form, and even how they're used in the real world. Solving y = 3x² - 12x + 16 might have seemed daunting at first, but hopefully, now you see it's totally manageable when you break it down step by step. Remember, the key to mastering quadratic equations (or any math topic, really) is practice. The more you work through problems, the more comfortable you'll become with the techniques and the better you'll understand the concepts. Don't be afraid to make mistakes – they're part of the learning process. And don't hesitate to review the steps, try different examples, and ask questions if you get stuck. Math is like building a tower – each concept builds upon the previous one. So, a solid understanding of quadratic equations will set you up for success in more advanced math topics. Keep practicing, stay curious, and you'll be solving quadratic equations like a pro in no time! You've got this!