Math question?

Question

If a + b = 2
and a^2 + b^2 = 3
then what is a^3 + b^3?
I got 3. Is that correct or logical?
Here is my reasoning:
since a + b =2, (a+b)^2 = 4 = a^2 + 2ab + b^2. a^2 + b^2 = 3, so if we plug in the numbers, 3+ 2ab = 4 = (a + b)^2. so 2ab=1, so ab= 0.5.
a^3 + b^3 = (a+b) (a - ab +b) = 2(2-ab)= 2(2-0.5) = 2 x 1.5 = 3.
I think a^3 + b^3=3.
Do you think this is correct?

Thank you Undertaker! The person who gave me the a^3 + b^3 formula told me the wrong one. I would have gotten the answer myself with the right formula, since you could see my reasoning was correct. Thank you for showing me my mistake!

Answer 1

a + b = 2.....(1)
a^2 + b^2 = 3.....(2)

Square both sides of (1)
(a + b)^2 = 4
a^2 + b^2 + 2ab = 4
2ab + 3 = 4 (From (2))
2ab = 1
ab = 1/2.....(3)

Do you know how this can be used?
There is an identity:
a^3 + b^3 = (a + b)(a^2 + b^2 - ab)

Using (1), (2) and (3), find a^3 + b^3

a^3 + b^3 = (a + b)(a^2 + b^2 - ab)
= 2(3 - 1/2)
= 6 - 1
= 5

a^3 + b^3 = 5