#### Question

If x,y are real numbers such that x^4+x^2=3,y^4-y^2=3．

Find the value of x^4+y^4．

Question

If x,y are real numbers such that x^4+x^2=3,y^4-y^2=3．

Find the value of x^4+y^4．

Given

x^4+x^2=3

(1)

y^4-y^2=3

(2)

Subtract (2) from (1)

x^4+x^2-(y^4-y^2) =0

Group terms for factorization

x^4-y^4+(x^2+y^2) = 0

(x^2+y^2)(x^2-y^2+1)=0

From given conditions (1), (2), x,y\ne 0

Therefore

x^2-y^2+1 = 0

x^2-y^2 = -1

(3)

Addition of (1) and (2) gives the equation

x^4+x^2+y^4-y^2 = 6

Then

x^4+y^4 = 6-(x^2-y^2)

Substitute value of x^2-y^2 given in (3)

x^4+y^4 = 6-(-1) = 7