The real number x and y satisfy
x + y = 4
x^2 + y^2 = 22
x^4 = y^4 - 176√7
+128475 x + y = 4
x^2 + y^2 =22
x^4 = y^4 - 176√7
Rearranging the last equation we have that
x^4 - y^4 = -176√7 factor
(x^2 - y^2) ( x^2 + y^2) = -176√7
(x + y) ( x - y) ( x^2 + y^2) = -176√7 subbinng into this we have that
(4) (x - y) ( 22) = -176√7
88 ( x - y) = -176√7 divide both sides by 88
x - y = -2√7
So
x + y = 4
x - y = -2√7 add these
2x = 4 - 2√7 divide both sides by 2
x = 2 - √7
And
( 2 - √7) + y = 4
y = 4 - (2 - √7)
y = 2 + √7
