This is an identity, verify yourself
a^3+b^3=(a+b)(a^2-ab+b^2)

substituting the values to reduce insanity
3=(1)(a^2-ab+b^2)
a^2+b^2=3+ab

Now we need ab.

(a+b)^3=a^3+3a^2b+3ab^2+b^3
(a+b)^3=a^3+b^3+3ab(a+b)
(1)^3=3+3ab(1)
1=3+3ab
ab=-2/3

a^2+b^2=3+(-2/3)
=3-2/3
=7/3

Or there abouts. You should check my math because I cannot decipher all these hat symbols. Who are I