Finding Separate Equations Of X² - 4y² - 2x + 4y = 0 A Step By Step Guide

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Let's embark on a mathematical journey to dissect the equation x² - 4y² - 2x + 4y = 0 and unveil its separate equations. This equation, at first glance, might seem like a jumbled mess of variables and coefficients, but with a bit of algebraic manipulation, we can expose its hidden structure. Our goal is to transform this single equation into a set of simpler equations that represent the same geometric figure. This process not only simplifies the equation but also provides deeper insights into the nature of the curve it represents. We will employ techniques such as completing the square and factoring to achieve this separation. By the end of this exploration, you'll not only understand how to find the separate equations but also appreciate the underlying mathematical principles that make it possible. We'll break down each step, ensuring clarity and comprehension, so even if you're not a mathematical whiz, you'll be able to follow along and grasp the concepts. This exercise is not just about solving an equation; it's about developing problem-solving skills that are applicable across various mathematical domains. So, let's dive in and unravel the mysteries of this equation!

Step 1: Rearranging and Grouping Terms

Our initial equation is x² - 4y² - 2x + 4y = 0. The first step towards unraveling this equation is to strategically rearrange and group the terms. This allows us to identify potential patterns and structures that might not be immediately obvious. Specifically, we aim to group the x-terms together and the y-terms together. This grouping sets the stage for applying techniques like completing the square, which is crucial for simplifying quadratic expressions. By bringing similar terms together, we create a more organized and manageable expression, making it easier to identify opportunities for factorization or other algebraic manipulations. Think of it like organizing your tools before starting a project – it makes the entire process smoother and more efficient. So, let's proceed by grouping the x-terms (x² and -2x) and the y-terms (-4y² and +4y) together, laying the foundation for the next stage of our mathematical exploration. This seemingly simple rearrangement is a powerful technique that can transform complex equations into more tractable forms. This step is essential for revealing the underlying structure of the equation and paving the way for further simplification.

Therefore, we can rewrite the equation as: (x² - 2x) - (4y² - 4y) = 0

Step 2: Completing the Square

Now, the heart of our equation-solving strategy lies in a powerful technique called completing the square. This method allows us to transform quadratic expressions into perfect square trinomials, which are much easier to factor and manipulate. For the x-terms (x² - 2x), we need to add and subtract a constant term that will create a perfect square. The constant we need is (b/2)², where 'b' is the coefficient of the x term. In this case, b = -2, so (b/2)² = (-2/2)² = 1. Similarly, for the y-terms (4y² - 4y), we first factor out the coefficient of y², which is 4, resulting in 4(y² - y). Then, we complete the square inside the parentheses. Here, the coefficient of the y term is -1, so the constant we need to add and subtract is (-1/2)² = 1/4. Remember, completing the square is not just a mechanical process; it's about transforming the equation into a form that reveals its underlying structure. It's like adding the missing piece to a puzzle, allowing the bigger picture to emerge. By completing the square, we pave the way for factoring the equation and ultimately finding its separate equations. This step is a crucial turning point in our solution, transforming the equation into a more manageable form that exposes its hidden properties.

For the x-terms:

  • We have x² - 2x.
  • To complete the square, we add and subtract (2/2)² = 1.
  • This gives us (x² - 2x + 1) - 1, which can be written as (x - 1)² - 1.

For the y-terms:

  • We have 4y² - 4y = 4(y² - y).
  • To complete the square inside the parentheses, we add and subtract (1/2)² = 1/4.
  • This gives us 4[(y² - y + 1/4) - 1/4], which can be written as 4[(y - 1/2)² - 1/4].
  • Distributing the 4, we get 4(y - 1/2)² - 1.

Step 3: Substituting Back into the Equation

Now that we've masterfully completed the square for both the x and y terms, the next crucial step is to substitute these expressions back into our original equation. This substitution is a pivotal moment in our solution, as it brings together the results of our previous manipulations and allows us to see the equation in a new light. By replacing the original quadratic expressions with their completed square forms, we transform the equation into a form that is much more conducive to factoring. It's like swapping out complex machinery for simpler components, making the overall system easier to understand and work with. This step is not just a mechanical substitution; it's about synthesizing our previous work and creating a new, more revealing representation of the equation. It sets the stage for the final act of our solution, where we'll factor the equation and unveil its separate equations. This substitution is the bridge that connects our individual efforts on the x and y terms, leading us towards the ultimate goal of separating the equation.

Substituting (x - 1)² - 1 for (x² - 2x) and 4(y - 1/2)² - 1 for -(4y² - 4y), we get:

(x - 1)² - 1 - [4(y - 1/2)² - 1] = 0

Step 4: Simplifying the Equation

With our substitutions in place, it's time to embark on a simplification journey. This involves carefully distributing any negative signs, combining like terms, and generally tidying up the equation to make it more manageable. Simplification is a crucial skill in mathematics, as it allows us to reduce complex expressions to their most basic forms, revealing their underlying structure and making them easier to work with. In this case, we need to pay close attention to the negative sign in front of the bracketed y-terms, ensuring that it is properly distributed to each term inside. This process of simplification is like cleaning up a cluttered workspace – it removes distractions and allows us to focus on the essential elements. By simplifying the equation, we prepare it for the next critical step: factoring. A simplified equation is not just aesthetically pleasing; it's also a more powerful tool for mathematical analysis. This step is essential for revealing the hidden relationships within the equation and paving the way for a clear and concise solution.

Expanding and simplifying, we have:

(x - 1)² - 1 - 4(y - 1/2)² + 1 = 0

(x - 1)² - 4(y - 1/2)² = 0

Step 5: Factoring the Difference of Squares

Now comes the elegant application of a fundamental algebraic identity: the difference of squares. This identity states that a² - b² can be factored into (a + b)(a - b). Recognizing this pattern in our equation is a key insight that allows us to break it down into simpler components. In our case, we can view (x - 1)² as a² and 4(y - 1/2)² as b². This identification is like finding the right key to unlock a door, revealing the path to the solution. Factoring is a cornerstone of algebraic manipulation, and the difference of squares is one of the most frequently encountered and useful factoring patterns. By applying this identity, we transform a seemingly complex equation into a product of two simpler expressions, each of which represents a linear equation. This step is not just a mechanical application of a formula; it's about recognizing and exploiting a fundamental mathematical structure. It's a testament to the power of algebraic identities in simplifying and solving equations. Factoring the difference of squares is a pivotal step that brings us closer to unveiling the separate equations of our original expression.

We can rewrite the equation as:

(x - 1)² - [2(y - 1/2)]² = 0

Applying the difference of squares factorization, a² - b² = (a + b)(a - b), where a = (x - 1) and b = 2(y - 1/2), we get:

[(x - 1) + 2(y - 1/2)][(x - 1) - 2(y - 1/2)] = 0

Step 6: Deriving the Separate Equations

We've arrived at the final stage of our mathematical journey: deriving the separate equations. The factored form of our equation, [(x - 1) + 2(y - 1/2)][(x - 1) - 2(y - 1/2)] = 0, tells us that the product of two expressions is zero. This fundamental principle of algebra dictates that at least one of these expressions must be equal to zero. This is the key that unlocks the separate equations. Each expression represents a linear equation, and setting each one equal to zero gives us the individual equations that make up the original equation. This step is not just about solving for variables; it's about understanding the relationship between the factored form and the individual equations it represents. It's the culmination of all our previous efforts, bringing together the techniques of completing the square, factoring, and algebraic manipulation. By deriving the separate equations, we gain a complete understanding of the geometric figure represented by the original equation. This is the ultimate goal of our exploration – to transform a complex equation into its simplest components, revealing its true nature and structure.

Now, we set each factor equal to zero:

  1. (x - 1) + 2(y - 1/2) = 0

    Simplifying, we get:

    x - 1 + 2y - 1 = 0

    x + 2y - 2 = 0

    x + 2y = 2

  2. (x - 1) - 2(y - 1/2) = 0

    Simplifying, we get:

    x - 1 - 2y + 1 = 0

    x - 2y = 0

Conclusion

In conclusion, we've successfully navigated the mathematical landscape to find the separate equations of x² - 4y² - 2x + 4y = 0. Through a series of strategic steps – rearranging terms, completing the square, substituting back into the equation, simplifying, and factoring – we've transformed a complex equation into its fundamental components. The separate equations we've derived, x + 2y = 2 and x - 2y = 0, represent two lines that, when considered together, form the geometric figure described by the original equation. This process has not only demonstrated the power of algebraic manipulation but also highlighted the importance of recognizing patterns and applying fundamental mathematical principles. The journey from the initial equation to the separate equations is a testament to the beauty and elegance of mathematics, where complex problems can be broken down into simpler, more manageable steps. This exercise serves as a valuable lesson in problem-solving, applicable not only in mathematics but also in various other domains. By mastering techniques like completing the square and factoring, we equip ourselves with powerful tools for tackling a wide range of mathematical challenges. The separate equations, in their simplicity, reveal the underlying structure of the original equation, providing a deeper understanding of its nature and properties. This is the essence of mathematical exploration – to uncover hidden relationships and reveal the beauty of mathematical structures.

Therefore, the separate equations are:

  • x + 2y = 2
  • x - 2y = 0