Question

X^2+ 2x = y^2 - 2y + 1993

Answer 1

вот) должно быть правильно

Answer 2

The equation you've provided is a quadratic equation involving both x and y variables. It can be rearranged to bring all terms to one side and simplify it:

x^2 + 2x - y^2 + 2y = 1993

To further work with this equation, we can complete the square separately for the x and y variables:

For the x terms: x^2 + 2x = x^2 + 2x + 1 - 1 = (x + 1)^2 - 1

For the y terms: -y^2 + 2y = -(y^2 - 2y) = -(y^2 - 2y + 1 - 1) = -((y - 1)^2 - 1) = - (y - 1)^2 + 1

Substituting these simplified expressions back into the main equation:

(x + 1)^2 - 1 - (y - 1)^2 + 1 = 1993

Simplify the equation:

(x + 1)^2 - (y - 1)^2 = 1993

Now, this equation represents a difference of squares on the left-hand side. We can use the difference of squares formula, which states that a^2 - b^2 = (a + b)(a - b):

[(x + 1) + (y - 1)][(x + 1) - (y - 1)] = 1993

Simplify further:

(x + 1 + y - 1)(x + 1 - y + 1) = 1993 (x + y)(x - y + 2) = 1993

So, the equation in factored form is:

(x + y)(x - y + 2) = 1993

To proceed, you might want to find the factors of 1993 and try to solve for x and y using these factor pairs. However, please note that 1993 is a prime number, which means it only has two factors: 1 and itself. This makes solving for integer solutions of x and y more challenging, and it might not have any integer solutions.