How To Convert The Equation $-2y^2 + X - 4y + 6 = 0$ Into Standard Form?

In mathematics, converting equations into their standard form is a fundamental skill. This process not only simplifies the equation but also reveals key information about the represented curve or shape. In this article, we'll delve into a specific example: converting the equation $-2y^2 + x - 4y + 6 = 0$ into its standard form. This is a crucial technique in algebra and analytic geometry, enabling us to easily identify the properties of the conic section represented by the equation. Understanding standard forms allows for quick recognition of the type of curve (parabola, ellipse, hyperbola) and its essential features, such as the vertex, axis of symmetry, and direction of opening. Our focus here will be on transforming the given equation into a form that highlights these properties, making it simpler to analyze and graph.

Understanding the Importance of Standard Form

Before we dive into the conversion process, it's important to understand why the standard form is so valuable. The standard form of a quadratic equation provides a clear and concise representation of the equation, making it easier to analyze and interpret. In our case, the standard form will allow us to readily identify the vertex, axis of symmetry, and direction of the parabola represented by the equation. By rearranging the terms and completing the square, we can rewrite the given equation into a form that directly reveals these characteristics. This transformation is not just about algebraic manipulation; it's about gaining a deeper understanding of the equation's geometric meaning. The standard form acts as a key, unlocking the essential properties of the curve it represents, and allowing us to visualize and manipulate it with greater ease. This concept is not only critical in the context of mathematical problem-solving but also has applications in various fields such as physics, engineering, and computer graphics, where understanding and manipulating curves and shapes is paramount.

Problem Statement

Our objective is to transform the given equation, $-2y^2 + x - 4y + 6 = 0$, into its standard form. This means we need to manipulate the equation algebraically to isolate x on one side and express the other side as a quadratic expression in terms of y. Specifically, we are aiming for a form that resembles x = a(y - k)² + h, where (h, k) represents the vertex of the parabola. This standard form provides immediate insight into the parabola's properties. The coefficient a dictates the direction and width of the opening, while the vertex (h, k) pinpoints the parabola's extreme point. The process of converting to standard form involves completing the square, a technique used to rewrite quadratic expressions in a more revealing format. By applying this technique, we can systematically rearrange the terms, factor out coefficients, and add appropriate constants to both sides of the equation, ultimately achieving the desired standard form. This process not only solves the specific problem at hand but also reinforces a fundamental algebraic skill applicable in a wide range of mathematical contexts.

Step-by-Step Solution

1. Isolate x:

Our first step is to isolate x on one side of the equation. This is achieved by moving all other terms to the opposite side. Starting with the given equation: $-2y^2 + x - 4y + 6 = 0$, we add $2y^2$, $4y$, and subtract $6$ from both sides. This process maintains the equation's balance while focusing our attention on expressing x in terms of y. The isolation of x is a critical initial step because it sets the stage for the subsequent steps involving completing the square. By isolating x, we're essentially setting up the equation to be expressed in a form that highlights the parabolic relationship between x and y. This step not only simplifies the equation but also provides a clear direction for the next stages of the conversion process. The act of isolating a variable is a fundamental algebraic technique, applicable in a wide range of problem-solving scenarios, making this step a crucial component of our solution.

x = 2y^2 + 4y - 6

2. Factor out the coefficient of $y^2$:

Now, we focus on the right side of the equation. The next step involves factoring out the coefficient of the $y^2$ term, which in this case is 2. Factoring out this coefficient is essential for completing the square, as it ensures that the quadratic expression inside the parentheses has a leading coefficient of 1. This simplifies the process of finding the constant term needed to complete the square. The step of factoring is not merely a mechanical procedure; it's a strategic move that simplifies the subsequent algebraic manipulations. By removing the coefficient from the $y^2$ term, we create a more manageable quadratic expression within the parentheses, setting the stage for the critical step of completing the square. This technique is a cornerstone of quadratic equation manipulation and is applicable in various contexts beyond this specific problem.

x = 2(y^2 + 2y) - 6

3. Complete the square:

This is the core step in converting the equation to standard form. To complete the square, we take half of the coefficient of the y term (which is 2), square it (which gives us 1), and add it inside the parentheses. However, since we're adding it inside the parentheses that are being multiplied by 2, we must also account for this by effectively adding 2 * 1 = 2 to the right side of the equation. Completing the square is a technique that transforms a quadratic expression into a perfect square trinomial, which can then be factored into a squared binomial. This process is crucial for rewriting the equation in standard form, as it allows us to express the relationship between x and y in a way that reveals the vertex of the parabola. The act of completing the square is not just a mathematical trick; it's a powerful technique that unlocks the hidden structure within the quadratic expression, making it easier to analyze and interpret.

x = 2(y^2 + 2y + 1) - 6 - 2(1)
x = 2(y^2 + 2y + 1) - 6 - 2

4. Factor the perfect square trinomial:

The expression inside the parentheses, $y^2 + 2y + 1$, is now a perfect square trinomial. This can be factored into $(y + 1)^2$. Factoring the perfect square trinomial is the culmination of the completing the square process. This step condenses the quadratic expression into a squared binomial, which is a key component of the standard form equation. The ability to recognize and factor perfect square trinomials is a valuable skill in algebra, as it simplifies expressions and reveals underlying structures. In this context, factoring the trinomial allows us to express the equation in a form that directly relates x to the squared term (y + 1)², paving the way for the final simplification and presentation of the standard form.

x = 2(y + 1)^2 - 8

5. Simplify the equation:

Finally, we simplify the equation by combining the constant terms. This step ensures that the equation is presented in its most concise and readily interpretable form. Simplifying the equation is not merely a cosmetic step; it's about clarity and precision. By combining the constant terms, we arrive at the final standard form equation, which clearly reveals the parabola's vertex and other key properties. This step underscores the importance of attention to detail in mathematical problem-solving, as even seemingly minor simplifications can significantly enhance understanding and facilitate further analysis.

x = 2(y + 1)^2 - 8

Final Answer

Therefore, the standard form of the equation $-2y^2 + x - 4y + 6 = 0$ is:

B. x = 2(y + 1)² - 8

Conclusion

Converting the quadratic equation $-2y^2 + x - 4y + 6 = 0$ into standard form, $x = 2(y + 1)^2 - 8$, is a testament to the power of algebraic manipulation. By isolating x, factoring out the coefficient of $y^2$, completing the square, and simplifying, we've unveiled the underlying structure of the equation. This standard form not only provides a concise representation but also immediately reveals the vertex of the parabola at (-8, -1) and its direction of opening. The process highlights the importance of mastering algebraic techniques like completing the square, which are fundamental tools in various mathematical contexts. Furthermore, understanding standard forms is crucial for recognizing and analyzing conic sections, allowing for efficient problem-solving and a deeper appreciation of the relationships between equations and their graphical representations. This conversion process serves as a valuable exercise in algebraic thinking and reinforces the connection between mathematical equations and the geometric shapes they describe.