Alright, let's dive into converting the quadratic equation y = 3x^2 + 30x + 82 into vertex form. This is a common task in algebra, and understanding how to do it can really boost your problem-solving skills. Vertex form not only makes it easier to identify the vertex of the parabola but also provides insights into the graph's transformations. So, grab your pencils, and let’s get started!

Understanding Vertex Form

Before we jump into the conversion, let's quickly recap what vertex form actually is. The vertex form of a quadratic equation is given by:

y = a(x - h)^2 + k

Where:

• (h, k) is the vertex of the parabola.
• a is the same leading coefficient as in the standard form (y = ax^2 + bx + c), which determines whether the parabola opens upwards (if a > 0) or downwards (if a < 0), and also affects its width.

Converting to vertex form involves completing the square. This technique allows us to rewrite the quadratic equation in a way that reveals the vertex directly. It might seem a bit tricky at first, but with practice, it becomes second nature. The vertex, (h, k), is a crucial point on the parabola, representing its minimum or maximum value. Identifying this point is incredibly useful in various applications, from optimization problems to graphing. Understanding the role of 'a' is also vital. If 'a' is positive, the parabola opens upwards, indicating a minimum value at the vertex. Conversely, if 'a' is negative, the parabola opens downwards, indicating a maximum value at the vertex. The magnitude of 'a' affects how stretched or compressed the parabola is; a larger absolute value of 'a' results in a narrower parabola, while a smaller absolute value results in a wider one.

Step-by-Step Conversion

Now, let's convert the given equation y = 3x^2 + 30x + 82 into vertex form step-by-step.

Step 1: Factor out the Leading Coefficient

First, we need to factor out the leading coefficient (which is 3 in this case) from the x^2 and x terms. This gives us:

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

Factoring out the leading coefficient is a critical first step because it allows us to focus on completing the square for the quadratic expression inside the parenthesis. This ensures that we're working with a standard form that's easier to manipulate. By isolating the x^2 and x terms, we set the stage for creating a perfect square trinomial, which is essential for converting the equation into vertex form. Remember, we only factor out the coefficient from the x^2 and x terms, leaving the constant term (82 in this case) untouched for now. This maintains the balance of the equation and prepares us for the next step in the conversion process. It's like preparing your ingredients before you start cooking – getting everything in place makes the rest of the process smoother and more efficient.

Step 2: Complete the Square

To complete the square for the expression inside the parenthesis (x^2 + 10x), we need to add and subtract a value that makes it a perfect square trinomial. This value is (b/2)^2, where b is the coefficient of the x term. In this case, b = 10, so (b/2)^2 = (10/2)^2 = 5^2 = 25. Thus:

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

Completing the square involves transforming a quadratic expression into a perfect square trinomial, which can then be easily factored into a squared term. This transformation is crucial for converting the quadratic equation into vertex form because it allows us to identify the vertex of the parabola. The process hinges on adding and subtracting a specific value, calculated as (b/2)^2, inside the parenthesis. This value ensures that the resulting trinomial can be factored into the form (x + b/2)^2. By adding and subtracting the same value, we maintain the equation's balance while simultaneously creating the perfect square trinomial. This step might seem a bit abstract, but it's a fundamental technique in algebra that has wide-ranging applications beyond just converting to vertex form. It's like finding the missing piece of a puzzle that, once fitted, reveals a clearer picture of the equation's structure.

Step 3: Rewrite as a Perfect Square Trinomial

Now, we rewrite the perfect square trinomial:

y = 3((x + 5)^2 - 25) + 82

Rewriting the expression as a perfect square trinomial is the heart of the completing the square method. By recognizing that x^2 + 10x + 25 can be factored into (x + 5)^2, we simplify the equation and bring it closer to vertex form. This step transforms the quadratic expression from a sum of individual terms into a compact, squared term, which directly reveals the x-coordinate of the vertex. The squared term (x + 5)^2 represents a parabola that has been shifted horizontally. The value inside the parenthesis, in this case, +5, indicates the direction and magnitude of the shift. It's like repackaging a complex set of instructions into a single, easily understandable command. This simplification not only makes the equation more manageable but also provides valuable insight into the parabola's behavior and position on the coordinate plane. It's a key step in unlocking the secrets of the quadratic equation.

Step 4: Distribute and Simplify

Distribute the 3 and simplify:

y = 3(x + 5)^2 - 75 + 82 y = 3(x + 5)^2 + 7

Distributing the leading coefficient and simplifying the equation is a crucial step in isolating the vertex form. By multiplying the 3 back into the parenthesis, we ensure that the equation remains balanced and that all terms are properly accounted for. This distribution affects the constant term outside the parenthesis, which ultimately determines the y-coordinate of the vertex. Simplifying the equation involves combining the constant terms to arrive at a single, consolidated value. This step tidies up the equation and presents it in its final vertex form, making it easy to identify the vertex coordinates. It's like cleaning up your workspace after completing a task – organizing the elements to reveal the finished product in its clearest form. This simplification not only makes the equation more visually appealing but also prepares it for easy interpretation and application in various contexts.

The Vertex Form

So, the vertex form of the equation y = 3x^2 + 30x + 82 is:

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

From this form, we can easily identify the vertex of the parabola as (-5, 7). The vertex form provides a concise and informative representation of the quadratic equation, highlighting its key features and making it easier to analyze and manipulate. This form reveals the vertex coordinates directly, allowing for quick identification of the parabola's minimum or maximum point. Additionally, the leading coefficient remains visible, indicating the parabola's direction and width. The vertex form is particularly useful in graphing the parabola, solving optimization problems, and understanding transformations. It's like having a blueprint that outlines the essential characteristics of the quadratic equation, enabling efficient analysis and problem-solving. Mastering the conversion to vertex form is a valuable skill that empowers you to tackle a wide range of algebraic challenges.

Conclusion

And there you have it! We've successfully converted the quadratic equation y = 3x^2 + 30x + 82 into vertex form, which is y = 3(x + 5)^2 + 7. Remember, guys, the key to mastering these conversions is practice. Keep at it, and you'll become a pro in no time!

Converting quadratic equations into vertex form is a fundamental skill in algebra with numerous applications. By understanding the steps involved in completing the square and recognizing the structure of vertex form, you can gain valuable insights into the behavior and properties of parabolas. This skill not only enhances your problem-solving abilities but also deepens your understanding of mathematical concepts. Practice converting various quadratic equations to vertex form, and you'll find yourself becoming more confident and proficient in algebra. Remember, the journey of mastering mathematics is a continuous process of learning, practicing, and applying what you've learned. So, keep exploring, keep practicing, and keep pushing your boundaries!