Hey guys! Today, we're diving into the world of quadratic equations and focusing on converting one into its vertex form. Specifically, we're tackling the equation y = 3x^2 + 30x + 82. Understanding how to convert a quadratic equation to vertex form is super useful in algebra, as it helps us quickly identify the vertex of the parabola, which is a critical point for understanding the function's behavior. The vertex form of a quadratic equation is given by y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. Our mission is to rewrite the given equation in this format. So, let's break it down step by step to make sure everyone's on board and can follow along.

Step-by-Step Conversion

1. Factor out the 'a' value

The first thing we need to do is factor out the coefficient of the x^2 term, which is '3' in our case, from the first two terms of the equation. This prepares us for completing the square. Factoring '3' out of 3x^2 + 30x gives us 3(x^2 + 10x). So, our equation now looks like this:

y = 3(x^2 + 10x) + 82

Keeping that '3' outside the parenthesis is super important because it affects the entire expression inside. It's like setting the stage for the next act, which is all about completing the square.

2. Completing the Square

Completing the square is the heart of this conversion process. We need to turn the expression inside the parenthesis, x^2 + 10x, into a perfect square trinomial. A perfect square trinomial can be factored into the form (x + n)^2. To find the value that completes the square, we take half of the coefficient of our 'x' term (which is 10), square it, and add it inside the parenthesis. Half of 10 is 5, and 5 squared is 25. So, we add 25 inside the parenthesis:

y = 3(x^2 + 10x + 25) + 82

But hold on! We can't just add 25 inside the parenthesis without balancing the equation. Since everything inside the parenthesis is being multiplied by 3, we're actually adding 3 * 25 = 75 to the right side of the equation. To keep things balanced, we must subtract 75 outside the parenthesis:

y = 3(x^2 + 10x + 25) + 82 - 75

3. Factor and Simplify

Now, we can factor the perfect square trinomial inside the parenthesis. x^2 + 10x + 25 factors to (x + 5)^2. And we simplify the constant terms outside the parenthesis: 82 - 75 = 7. So, our equation becomes:

y = 3(x + 5)^2 + 7

And there you have it! We've successfully converted the equation into vertex form.

Identifying the Vertex

From the vertex form y = 3(x + 5)^2 + 7, we can easily identify the vertex of the parabola. Remember, the vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex. In our equation, we have (x + 5), which can be rewritten as (x - (-5)). Therefore, h = -5 and k = 7. So, the vertex of the parabola is (-5, 7).

This means the parabola reaches its minimum (or maximum, depending on the sign of 'a') at the point x = -5 and the value of the function at that point is y = 7. Knowing the vertex helps us understand the graph of the quadratic equation. Since 'a' is positive (a = 3), the parabola opens upwards, and the vertex is the minimum point.

Significance of Vertex Form

The vertex form is incredibly useful because it directly reveals the vertex of the parabola. This is super helpful for graphing quadratic equations and for solving optimization problems. For example, if you need to find the minimum cost or maximum profit in a business scenario, and that scenario can be modeled by a quadratic equation, finding the vertex can give you the answer directly. It's also useful in physics, where projectile motion can be modeled using quadratic equations, and the vertex represents the highest point the projectile reaches.

Moreover, the vertex form makes it easy to apply transformations to the graph of the quadratic equation. Changing the value of 'a' stretches or compresses the parabola vertically. Changing 'h' shifts the parabola horizontally, and changing 'k' shifts it vertically. Understanding these transformations can help you quickly visualize and analyze different quadratic equations.

Common Mistakes to Avoid

When converting to vertex form, there are a few common mistakes that students often make. One of the most frequent errors is forgetting to account for the 'a' value when completing the square. Remember, if you add a value inside the parenthesis, you're actually adding 'a' times that value to the equation, so you need to subtract the same amount outside the parenthesis to keep the equation balanced. Another common mistake is getting the signs mixed up when identifying the vertex. The vertex form is y = a(x - h)^2 + k, so the x-coordinate of the vertex is the opposite of the value inside the parenthesis. For example, if the equation is y = a(x + 3)^2 + k, the x-coordinate of the vertex is -3, not 3.

Also, double-check your arithmetic, especially when squaring numbers and adding/subtracting constants. A small arithmetic error can throw off the entire solution. It's always a good idea to review your steps and make sure everything adds up correctly.

Practice Problems

To solidify your understanding, try converting these quadratic equations into vertex form:

1. y = 2x^2 - 8x + 10
2. y = -x^2 + 6x - 5
3. y = 4x^2 + 16x + 19

Work through these problems step by step, and be sure to check your answers. Practice makes perfect, and the more you practice, the more comfortable you'll become with the process.

Conclusion

Converting quadratic equations into vertex form might seem daunting at first, but with a clear understanding of the steps involved, it becomes a manageable and even enjoyable task. Remember to factor out the 'a' value, complete the square carefully, and balance the equation. By mastering this technique, you'll gain valuable insights into the properties of quadratic equations and their graphs. So keep practicing, and you'll be a vertex form pro in no time! Keep up the great work, and let me know if you have any other questions. You got this!