x+y=1

(x+y)^2 = 1

x^2+y^2+2xy = 1

\because x^2+y^2=2

\therefore 2+2xy = 1

xy = -\dfrac{1}{2}

x^4+y^4 = (x^2+y^2)^2-2x^2y^2

= 2^2-2\cdotp ( -\dfrac{1}{2} ) ^2

= 4 - \dfrac{1}{2}

=\dfrac{7}{2}