Find Parabola Vertex Completing The Square Method

In mathematics, understanding the properties of parabolas is crucial, especially in fields like calculus and physics. One essential characteristic of a parabola is its vertex, which represents the point where the parabola changes direction. The vertex can be particularly insightful when solving optimization problems or analyzing physical trajectories. The method of completing the square is a powerful algebraic technique that allows us to rewrite quadratic equations into a form that reveals the vertex coordinates directly. This article delves into how to use the completing the square method to find the vertex of a parabola given its equation.

The method of completing the square is a technique used to rewrite a quadratic equation in the form of $ax^2 + bx + c = 0$ into the vertex form, which is $a(x - h)^2 + k = 0$, where $(h, k)$ represents the vertex of the parabola. This method involves algebraic manipulations to create a perfect square trinomial, which can then be factored into a binomial squared. The transformation not only aids in finding the vertex but also simplifies the process of solving quadratic equations. Before we dive into the specific equation, let's discuss the general process of completing the square. The general process of completing the square involves several steps that transform a standard quadratic equation into vertex form. First, if the coefficient of the $x^2$ term (the 'a' in $ax^2 + bx + c$) is not 1, you need to factor it out from the terms containing $x$. This step ensures that you can create a perfect square trinomial more easily. Next, focus on the remaining quadratic expression inside the parentheses. Take half of the coefficient of the $x$ term (the 'b' in $bx$), square it, and then add and subtract this value inside the parentheses. Adding and subtracting the same value doesn't change the overall equation but allows us to rewrite part of it as a perfect square. The perfect square trinomial can then be factored into a binomial squared, such as $(x + p)^2$ or $(x - p)^2$. The remaining terms are combined to give the constant term 'k' in the vertex form. Finally, rewrite the equation in the form $a(x - h)^2 + k$, which directly gives you the vertex coordinates $(h, k)$. This form makes it easy to identify the vertex and understand the parabola's orientation and shape.

Applying the Method to the Given Equation

Now, let's apply this method to the given equation: $x^2 + 10x - y + 15 = 0$. Our primary goal is to rewrite this equation in the vertex form to easily identify the vertex of the parabola. To begin, we need to isolate the terms involving $x$ on one side of the equation. This involves moving the $y$ and constant terms to the other side, which sets the stage for completing the square. Start by adding $y$ and subtracting 15 from both sides of the equation. This gives us $x^2 + 10x = y - 15$. This step is crucial as it separates the quadratic and linear terms in $x$ from the $y$ term, allowing us to focus on completing the square for the $x$ terms. The next step involves completing the square for the quadratic expression $x^2 + 10x$. To do this, we take half of the coefficient of the $x$ term, which is 10, and square it. Half of 10 is 5, and squaring 5 gives us 25. This value, 25, is what we need to add to both sides of the equation to complete the square. Adding 25 to both sides, we get $x^2 + 10x + 25 = y - 15 + 25$. This step transforms the left side of the equation into a perfect square trinomial, which can be easily factored. Now, we factor the left side and simplify the right side of the equation. The left side, $x^2 + 10x + 25$, is a perfect square trinomial and can be factored as $(x + 5)^2$. The right side simplifies to $y + 10$. Thus, our equation becomes $(x + 5)^2 = y + 10$. This form is very close to the vertex form of a parabola, which makes it straightforward to identify the vertex coordinates. To express the equation in the standard vertex form, we rearrange it to solve for $y$. Subtract 10 from both sides to get $y = (x + 5)^2 - 10$. This equation is now in the vertex form $y = a(x - h)^2 + k$, where $a = 1$, $h = -5$, and $k = -10$. From this, we can directly read off the vertex coordinates. The vertex of the parabola is the point $(h, k)$, which in this case is $(-5, -10)$. This is the point where the parabola changes direction, and it is a critical feature of the parabola's graph.

Conclusion

In summary, completing the square is an effective method for rewriting quadratic equations into vertex form, allowing us to easily identify the vertex of a parabola. By following the steps of isolating the $x$ terms, adding and subtracting the square of half the coefficient of the $x$ term, and factoring the perfect square trinomial, we can transform the equation into the form $y = a(x - h)^2 + k$. This form directly reveals the vertex coordinates as $(h, k)$. Applying this method to the equation $x^2 + 10x - y + 15 = 0$, we found that the vertex of the parabola is $(-5, -10)$. Understanding how to find the vertex is essential for analyzing and graphing parabolas, as well as for solving various mathematical and real-world problems involving quadratic functions. The vertex not only gives us the minimum or maximum point of the parabola but also provides insights into its symmetry and overall shape. Mastering this technique enhances one's ability to work with quadratic equations and parabolas effectively.

Complete the Square to Find the Vertex of This Parabola

Equation

$\begin{array}{c} x^2+10 x-y+15=0 \\ \text { ([?], [ ]) } \end{array}$

Discussion Category

Mathematics

Rewriting the Equation

To find the vertex of the parabola given by the equation $x^2 + 10x - y + 15 = 0$, we will use the method of completing the square. This technique allows us to rewrite the equation in vertex form, which is $y = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The vertex form makes it easy to identify the vertex coordinates directly. Before we dive into the specifics of this equation, let's discuss why understanding the vertex of a parabola is so important. The vertex of a parabola is a critical point that provides significant information about the quadratic function it represents. It is the point where the parabola changes direction, making it either the minimum or maximum value of the function. In practical applications, the vertex can help us solve optimization problems, such as finding the maximum height of a projectile or the minimum cost in a business model. Understanding the vertex also aids in graphing parabolas accurately, as it serves as a key reference point. Additionally, the vertex form of the equation, obtained through completing the square, makes it easier to analyze transformations of the parabola, such as shifts and stretches. Therefore, mastering the technique of finding the vertex is essential for anyone working with quadratic functions.

The first step in completing the square is to isolate the $y$ term on one side of the equation. We can do this by adding $y$ to both sides of the equation, which gives us $x^2 + 10x + 15 = y$. This step is crucial because it sets up the equation in a form where we can focus on completing the square for the $x$ terms. By isolating $y$, we ensure that the subsequent algebraic manipulations will transform the quadratic expression in $x$ into the desired vertex form. Now that we have the equation in this form, we can proceed with the process of completing the square. The next step involves manipulating the left side of the equation to create a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the square of a binomial, such as $(x + a)^2$ or $(x - a)^2$. In our case, we need to find a value to add to $x^2 + 10x$ that will make it a perfect square. To find this value, we take half of the coefficient of the $x$ term, which is 10, and square it. Half of 10 is 5, and 5 squared is 25. Therefore, we need to add 25 to complete the square. However, to maintain the equality of the equation, we must also subtract 25. Adding and subtracting the same value ensures that we are not changing the overall value of the expression, but rather rewriting it in a more convenient form. This technique is a fundamental aspect of completing the square and is essential for transforming the quadratic equation into vertex form. So, we add and subtract 25 to the left side of the equation, resulting in $x^2 + 10x + 25 - 25 + 15 = y$. This sets the stage for factoring the perfect square trinomial and simplifying the equation.

Completing the Square

Now we rewrite the left side by grouping the perfect square trinomial: $(x^2 + 10x + 25) - 25 + 15 = y$. The expression in the parentheses is a perfect square trinomial, which can be factored as $(x + 5)^2$. This is a key step in the process of completing the square, as it transforms the quadratic expression into a squared binomial. The ability to recognize and factor perfect square trinomials is crucial for efficiently solving quadratic equations and finding the vertex of parabolas. The factored form simplifies the equation and brings us closer to the vertex form of the parabola. After factoring, the equation becomes $(x + 5)^2 - 25 + 15 = y$. Next, we simplify the constant terms on the left side. Combining -25 and +15 gives us -10. So, the equation becomes $(x + 5)^2 - 10 = y$. This equation is now in the vertex form, $y = a(x - h)^2 + k$, where $a$ is the coefficient of the squared term, $h$ is the x-coordinate of the vertex, and $k$ is the y-coordinate of the vertex. In our case, $a = 1$, $h = -5$, and $k = -10$. The vertex form clearly reveals the vertex of the parabola, making it easy to identify its coordinates and understand the parabola's properties.

Identifying the Vertex

From the vertex form $y = (x + 5)^2 - 10$, we can directly identify the vertex of the parabola. The vertex is given by the coordinates $(h, k)$, where $h$ is the value that makes the term inside the square zero, and $k$ is the constant term. In this case, the term inside the square is $(x + 5)$, which becomes zero when $x = -5$. Therefore, $h = -5$. The constant term is -10, so $k = -10$. This gives us the vertex coordinates $(-5, -10)$. Understanding how to extract the vertex coordinates from the vertex form is essential for quickly analyzing parabolas and their properties. The vertex provides valuable information about the parabola's orientation, position, and extreme values. For example, if the coefficient $a$ is positive, the parabola opens upwards, and the vertex represents the minimum point. If $a$ is negative, the parabola opens downwards, and the vertex represents the maximum point. Additionally, the vertex is the axis of symmetry of the parabola, meaning that the parabola is symmetric about the vertical line passing through the vertex. Therefore, knowing the vertex allows us to easily sketch the graph of the parabola and understand its key characteristics.

Final Answer

Therefore, the vertex of the parabola given by the equation $x^2 + 10x - y + 15 = 0$ is $(-5, -10)$.

In conclusion, completing the square is a powerful technique for rewriting quadratic equations in vertex form, which makes it straightforward to identify the vertex of the parabola. By following the steps of isolating the $y$ term, adding and subtracting the appropriate value to complete the square, factoring the perfect square trinomial, and simplifying the equation, we can transform the equation into vertex form and extract the vertex coordinates. This method is essential for analyzing and graphing parabolas, as well as for solving various mathematical and real-world problems involving quadratic functions. The vertex provides valuable information about the parabola's minimum or maximum point, its symmetry, and its overall shape. Mastering this technique enhances one's ability to work with quadratic equations and parabolas effectively.