Find the value of x^2 + y^2

Let x+y=1 and x^3+y^3=19

Find the value of x^2 + y^2?

Answer

We have x^3 + y^3=(x+y)(x^2 - xy + y^2) = 19

and x+y = 1 Thus x^2 - xy + y^2 = 19

and also we can write as 2x^2 + 2y^2 - 2xy = 38(1)

Then also (x+y)^2 = x^2 + y^2 + 2xy = 1(2)

Adding (1) and (2):

We get 3x^2 + 3y^2 = 39

so that  x^2 + y^2 = 13