Solving the Equation (x^2 y^2 - 4x - 4y 8) to Determine (x - y)

When dealing with complex algebraic equations, it is essential to have a clear understanding of algebraic manipulations and completing the square. Let's walk through the process of solving the given equation:

The equation in question is: x^2 y^2 - 4x - 4y 8.

Step 1: Complete the Square for Both x and y

To complete the square, we need to rewrite the equation in a form that allows us to identify perfect squares on both x and y. Let's start with the given equation:

$$x^2 y^2 - 4x - 4y 8$$

First, let's handle the x terms:

$$x^2 - 4x$$

To complete the square, we add and subtract ((frac{-4}{2})^2 4).

$$(x^2 - 4x 4) - 4$$

Similarly, for the y terms:

$$y^2 - 4y$$

We add and subtract ((frac{-4}{2})^2 4).

$$(y^2 - 4y 4) - 4$$

Substituting these into the original equation, we get:

$$(x^2 - 4x 4) (y^2 - 4y 4) - 8 8$$

Simplifying, we get:

$$(x - 2)^2 (y - 2)^2 - 8 8$$ $$(x - 2)^2 (y - 2)^2 16$$

Step 2: Identify the Solutions for x and y

From the completed square form, we can see that:

$$(x - 2)^2 16 quad text{and} quad (y - 2)^2 16$$

Since the square of a real number can be 16 only if the number itself is 4 or -4, we get:

$$x - 2 4 quad text{or} quad x - 2 -4$$ $$y - 2 4 quad text{or} quad y - 2 -4$$

Solving for x and y, we get:

$$x 6 quad text{or} quad x -2$$ $$y 6 quad text{or} quad y -2$$

However, since we are looking for a specific value of (x - y), we need to check the pairs that satisfy both conditions:

$$(x, y) (6, 6), (-2, -2), (6, -2), (-2, 6)$$

For the pair ((6, 6)):

$$x - y 6 - 6 0$$

For the pair ((-2, -2)):

$$x - y -2 - (-2) 0$$

For the pairs ((6, -2)) and ((-2, 6)):

$$x - y 6 - (-2) 8 quad text{or} quad x - y -2 - 6 -8$$

Therefore, the value of (x - y) is:

$$x - y 0$$

Conclusion

From the analysis, we can see that the value of (x - y) is 0, as long as both (x) and (y) are either 6 or -2.

Thus, the final answer is:

$$ x - y 0$$

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