Question

For what values of x and y is the equation (x2 + y2)2 = (x2 – y2)2 + (2xy)2 true?

Answer 1

Answer

For any values of x and y is the equation(x2 + y2)2 = (x2 – y2)2 + (2xy)2  is true

Solution

Consider the identities

(a + b)^2 = a2 + b2 +2ab  ------(1)

(a - b)^2 = a2 + b2 -2ab   ------(2)

from (1) and (2) we can write

(a + b)^2 = (a - b)^2 + 4ab

Here a=x2 and b=y2

(x2 + y2)2  = (x2)2 + (y2)2 +2x2y2

(x2 - y2)2  = (x2)2 + (y2)2 -2x2y2

Therefore  

(x2 + y2)2 = (x2 – y2)2 + (2xy)2  

For any values of x and y is the equation (x2 + y2)2 = (x2 – y2)2 + (2xy)2  is true