%5E2%2B-4x-6y%2B7-%3D0.jpeg "(4x+2y-5)^2+ 4x-6y+7 =0")
Expanding and Simplifying the Equation (4x + 2y - 5)^2 + 4x - 6y + 7 = 0
The equation (4x + 2y - 5)^2 + 4x - 6y + 7 = 0 represents a quadratic equation in two variables, x and y. To analyze and work with this equation, we can simplify it by expanding and combining terms. Here's how we can do it:
1. Expanding the Square
First, we expand the squared term using the FOIL method or the perfect square trinomial formula:
(4x + 2y - 5)^2 = (4x + 2y - 5)(4x + 2y - 5)
= 16x² + 8xy - 20x + 8xy + 4y² - 10y - 20x - 10y + 25
= 16x² + 16xy + 4y² - 40x - 20y + 25
2. Combining Terms
Now, substitute the expanded term back into the original equation and combine like terms:
16x² + 16xy + 4y² - 40x - 20y + 25 + 4x - 6y + 7 = 0
This simplifies to:
16x² + 16xy + 4y² - 36x - 26y + 32 = 0
Analyzing the Simplified Equation
The simplified equation represents a conic section. Specifically, it's a generalized conic because it contains both x² and y² terms, as well as an xy term. This type of conic can represent various shapes like ellipses, hyperbolas, parabolas, or even degenerate cases (like a single point or a line).
Further steps to analyze the equation:
- Identify the type of conic: This can be done by examining the coefficients of the x², y², and xy terms. There are specific tests to determine the type of conic based on these coefficients.
- Find the center, focus (if applicable), and axes: These parameters help visualize and understand the specific conic represented by the equation.
- Graph the conic: This provides a visual representation of the equation's solution set.
Remember: Working with a generalized conic can be more complex than working with simpler conics like circles or parabolas.
This expanded and simplified equation provides a starting point for further analysis and understanding of the conic section represented.