X 2 Y 2 1 3 X 2y 3

Solving the Equation x² + 2y² + 1 = 3x + 2y + 3

This article explores how to solve the equation x² + 2y² + 1 = 3x + 2y + 3. We'll break down the process step-by-step, covering techniques for manipulating the equation and finding potential solutions. This problem combines elements of algebra and equation solving, making it a great exercise for practicing these fundamental mathematical skills.

The equation x² + 2y² + 1 = 3x + 2y + 3 appears deceptively simple, but requires a methodical approach to solve. Our goal is to find the values of x and y that satisfy this equation. This involves rearranging the equation into a more manageable form.

Rearranging the Equation

First, let's rearrange the equation to group similar terms together:

x² - 3x + 2y² - 2y + 1 - 3 = 0

Simplifying, we get:

x² - 3x + 2y² - 2y - 2 = 0

Now, we can employ the technique of completing the square for both the x and y terms. Completing the square allows us to transform a quadratic expression into a perfect square trinomial, making it easier to solve.

Completing the Square

Let's complete the square for the x terms:

x² - 3x can be rewritten as (x - 3/2)² - (3/2)² = (x - 3/2)² - 9/4

Similarly, for the y terms:

2y² - 2y can be rewritten as 2(y² - y). To complete the square, we take half of the coefficient of y (-1), square it (1/4), and add and subtract it inside the parentheses:

2(y² - y + 1/4 - 1/4) = 2((y - 1/2)² - 1/4) = 2(y - 1/2)² - 1/2

Substituting these back into our equation:

(x - 3/2)² - 9/4 + 2(y - 1/2)² - 1/2 - 2 = 0

Simplifying and Solving

Now, let's simplify and solve for x and y:

(x - 3/2)² + 2(y - 1/2)² = 9/4 + 1/2 + 2 = 9/4 + 5/4 = 14/4 = 7/2

This equation represents an ellipse. The center of the ellipse is at (3/2, 1/2). There are infinitely many pairs of (x, y) that satisfy this equation. To find specific solutions, you could choose a value for x and solve for y, or vice-versa. Numerical methods or graphing software could be utilized to find approximate solutions.

Conclusion

Solving x² + 2y² + 1 = 3x + 2y + 3 involves several steps, including rearranging the equation, completing the square, and simplifying to reveal the elliptical nature of the solution set. While we cannot find single unique solutions, we have successfully transformed the initial equation into a standard form that allows for graphical representation and the identification of an infinite number of solutions. Understanding these steps is crucial for mastering algebraic manipulation and solving similar quadratic equations.