Question

Знайдіть цілі розв’язки рівняння (х + у) 2 = (х + 2021)(у − 2021).​

Answer 1

Ответ:

2х+2у=ху-2021х+2021у - 4 084 441

2х+2у-ху+2021х-2021у + 4 084 441=0

-ху+2023х-2019у+4084441=0

Answer 2

Solving the Equation (x + y)^2 = (x + 2021)(y - 2021)

To find the solutions to the equation (x + y)^2 = (x + 2021)(y - 2021), we can start by expanding the left side of the equation using the binomial theorem:

(x + y)^2 = x^2 + 2xy + y^2

Now, let's expand the right side of the equation:

(x + 2021)(y - 2021) = xy - 2021x + 2021y - 2021^2

Combining like terms, we have:

x^2 + 2xy + y^2 = xy - 2021x + 2021y - 2021^2

Rearranging the terms, we get:

x^2 + y^2 - xy - 2021x - 2021y + 2021^2 = 0

Now, let's try to factorize the equation. However, it seems that the equation is not easily factorizable. Therefore, we will use an alternative method to solve it.

We can rewrite the equation as follows:

x^2 + y^2 - xy - 2021x - 2021y + 2021^2 = 0

Let's group the terms involving x and y:

(x^2 - xy - 2021x) + (y^2 - 2021y + 2021^2) = 0

Now, we can complete the square for both x and y separately.

For the x terms: (x^2 - xy - 2021x) = (x^2 - 2 * (x * (y/2)) - 2021x) = (x^2 - 2 * (xy/2) - 2021x) = (x^2 - 2 * (xy/2) - 2021x + (y/2)^2 - (y/2)^2) = (x^2 - 2 * (xy/2) - 2021x + (y/2)^2) - (y/2)^2 = (x - (y/2))^2 - (y/2)^2

For the y terms: (y^2 - 2021y + 2021^2) = (y^2 - 2 * (y * (2021/2)) + 2021^2) = (y^2 - 2 * (y * (2021/2)) + 2021^2 - (2021/2)^2) = (y^2 - 2 * (y * (2021/2)) + 2021^2 - (2021/2)^2) = (y - (2021/2))^2 - (2021/2)^2

Substituting these expressions back into the equation, we have:

(x - (y/2))^2 - (y/2)^2 + (y - (2021/2))^2 - (2021/2)^2 = 0

Simplifying further, we get:

(x - (y/2))^2 - (y/2)^2 + (y - (2021/2))^2 - (2021/2)^2 = 0

Now, we can rewrite the equation as:

(x - (y/2))^2 + (y - (2021/2))^2 = (y/2)^2 + (2021/2)^2

This equation represents the equation of a circle with center (y/2, 2021/2) and radius sqrt((y/2)^2 + (2021/2)^2).

Therefore, the equation (x + y)^2 = (x + 2021)(y - 2021) does not have a unique solution. Instead, it represents a family of circles with different centers and radii.

Please let me know if there's anything else I can help you with!