3. For real numbers $a$ and $b$, it holds that $a^{3}=3 a b^{2}+11$ and $b^{3}=3 a^{2} b+2$. Calculate the value of the expression $a^{2}+b^{2}$.
I/3. We rewrite the equations as $a^{3}-3 a b^{2}=11, b^{3}-3 a^{2} b=2$ and square them: $a^{6}-6 a^{4} b^{2}+9 a^{2} b^{4}=$ $121, b^{6}-6 a^{2} b^{4}+9 a^{4} b^{2}=4$. We add these two equations and get $125=a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}=$ $\left(a^{2}+b^{2}\right)^{3}$, so $a^{2}+b^{2}=5$.

Correct squaring of both equations

2 points

Sum of the squared equations

2 points

Rearrangement of terms with letter labels in $\left(a^{2}+b^{2}\right)^{3}$ 2 points

Conclusion $a^{2}+b^{2}=5$

1 point