Complete The Square: Find The Vertex Of Y=3x^2+15x+17
Hey guys! Today, we're going to dive into completing the square to find the vertex of the quadratic function: y = 3x^2 + 15x + 17. It might seem a bit daunting at first, but trust me, we'll break it down step by step so it's super easy to follow. Completing the square is an incredibly useful technique, not just for finding the vertex, but also for solving quadratic equations and understanding the properties of parabolas. So, grab your pencils, and let's get started!
Step 1: Factor out the Leading Coefficient
Our first mission is to factor out the leading coefficient, which in this case is 3, from the x^2 and x terms. This will help us get the quadratic expression into a form that's easier to complete the square. Factoring out the 3, we get:
y = 3(x^2 + 5x) + 17
Notice that we're only factoring the 3 out of the terms that contain x. The constant term, 17, stays outside the parentheses for now. This is a crucial step because it allows us to focus on completing the square within the parentheses without messing up the entire equation. By isolating the x^2 and x terms, we can manipulate them more easily to create a perfect square trinomial. Remember, the goal here is to rewrite the quadratic expression in vertex form, which will directly reveal the coordinates of the vertex. So, keep this factored form in mind as we move on to the next step. Remember, the goal is to rewrite this equation in vertex form.
Step 2: Complete the Square
Now comes the fun part – completing the square! To do this, we need to take half of the coefficient of our x term (which is 5), square it, and then add it inside the parentheses. Half of 5 is 5/2, and squaring that gives us (5/2)^2 = 25/4. So, we add 25/4 inside the parentheses. But here's the catch: since we're adding 25/4 inside the parentheses, which is being multiplied by 3, we're actually adding 3 * (25/4) = 75/4 to the equation. To keep the equation balanced, we need to subtract 75/4 outside the parentheses.
So, our equation becomes:
y = 3(x^2 + 5x + 25/4) + 17 - 75/4
Breaking this down, adding 25/4 inside the parenthesis allows us to rewrite the expression inside the parenthesis as a perfect square. Because we're not just adding 25/4 to the equation, we have to also compensate for this addition outside of the parenthesis. The compensation is required to maintain the equality of the equation, ensuring that we're not changing the fundamental relationship between x and y. The perfect square trinomial we've created is now easily factorable, which is the key to getting the equation into vertex form. Also, don't be intimidated by the fractions, guys! We'll deal with them in the next step. What's important now is understanding the logic behind adding and subtracting the term to complete the square.
Step 3: Rewrite as a Perfect Square and Simplify
Okay, let's rewrite the expression inside the parentheses as a perfect square. The expression x^2 + 5x + 25/4 is equivalent to (x + 5/2)^2. Now, let's simplify the constant terms outside the parentheses.
Our equation now looks like this:
y = 3(x + 5/2)^2 + 17 - 75/4
To combine 17 and -75/4, we need a common denominator. We can rewrite 17 as 68/4. So, we have:
y = 3(x + 5/2)^2 + 68/4 - 75/4
y = 3(x + 5/2)^2 - 7/4
And there you have it! We've successfully rewritten the quadratic function in vertex form. This form is super helpful because it directly tells us the coordinates of the vertex.
The vertex form is y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In our case, we can see that h = -5/2 and k = -7/4. So, the vertex of our parabola is (-5/2, -7/4). The transformation to vertex form wasn't just for show, guys! It's a strategic move that unlocks the key information about the parabola's turning point. You can now easily visualize the graph of the parabola and understand its behavior. Keep practicing these kinds of problems, and it'll be second nature in no time.
Step 4: Identify the Vertex
Now that we have the equation in vertex form, identifying the vertex is a piece of cake! Remember that the vertex form of a quadratic equation is given by:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex. Comparing this with our equation, y = 3(x + 5/2)^2 - 7/4, we can see that:
- h = -5/2
- k = -7/4
Therefore, the vertex of the parabola is located at the point (-5/2, -7/4). This is the point where the parabola changes direction, either reaching its minimum (if a > 0) or its maximum (if a < 0). In our case, a = 3, which is positive, so the parabola opens upwards, and the vertex represents the minimum point. Understanding how to extract the vertex from the vertex form of a quadratic equation is a fundamental skill in algebra and calculus. It allows you to quickly analyze the behavior of quadratic functions and solve a variety of problems related to optimization and curve sketching.
Conclusion
So, to recap, we started with the quadratic function y = 3x^2 + 15x + 17, and we used the method of completing the square to rewrite it in vertex form: y = 3(x + 5/2)^2 - 7/4. From this, we easily identified the vertex of the parabola as (-5/2, -7/4).
Completing the square might seem tricky at first, but with practice, it becomes a valuable tool in your mathematical toolkit. It's not just about finding the vertex; it's about understanding the structure of quadratic equations and how to manipulate them to reveal important information. Keep practicing, and you'll become a pro in no time! Remember, math is all about practice and patience, so don't get discouraged if you don't get it right away. Just keep at it, and you'll eventually master it. Now go out there and conquer those quadratic equations!