To solve for the standard form of the equation x^2 y^2 + 10x - 18y + 42 = 0, we can follow these steps:
1. Group the x and y terms separately.
x^2 y^2 + 10x - 18y + 42 = 0
(x^2 y^2 + 10x) + (-18y + 42) = 0
2. Complete the square for the x terms.
Take half of the coefficient of x (which is 10), square it (which is 25), and add it to both sides of the equation.
(x^2 y^2 + 10x + 25) + (-18y + 42) = 25
(x^2 y^2 + 10x + 25) = -(-18y + 42) + 25
3. Complete the square for the y terms.
Take half of the coefficient of y (which is -18), square it (which is 81), and add it to both sides of the equation.
(x^2 y^2 + 10x + 25) + (-18y + 42 + 81) = 25 + 81
(x^2 y^2 + 10x + 25) + (-18y + 123) = 106
4. Rewrite the equation in standard form.
(x^2 + 10x + 25) + (y^2 - 18y + 123) = 106
Now the equation is in standard form. Let me know if there's anything else I can help with!