If $a^3 + 12ab^2 = 679$ and $9a^2b +12b^3 = 978$, what is $a-2b$?

Question

If $a^3 + 12ab^2 = 679, 9a^2b +12b^3 = 978$ what is $a-2b$?

I tried adding them and factoring, subtracting them and factoring. I played with the equations but i couldn't solve them. Any solution is appreciated

Answer 1

\begin{align*} (a-2b)^3&=a^3+12ab^2-(8b^3+6a^2b)\\ &=679-\frac{2}{3}(12b^3+9a^2b)\\ &=679-\frac{2}{3}(978)\\ &=27\\ (a-2b)&=3. \end{align*}

Answer 2

If you're allowed to assume $a,b$ are integers, then $a$ divides $679=7\times97$, so try $a=7$; then $a^3+12ab^2=679$ gives $b^2=4$, and you find that $a=7$, $b=2$ works, so $a-2b=3$.