Bunuel wrote:
What is a–b?
(1) a^3 – b^3 = 117
(2) a^2 + ab + b^2 = 39
\(? = a - b\)
\(\left( 1 \right)\,\,{a^3} - {b^3} = 117\,\,\,\left\{ \begin{gathered}
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {\sqrt[{3\,}]{{117}}\,\,;\,\,0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = \sqrt[{3\,}]{{117}}\,\,\, \hfill \\
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {1\,\,;\,\, - \sqrt[{3\,}]{{116}}} \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = 1 + \sqrt[{3\,}]{{116}}\,\,\, \ne \sqrt[{3\,}]{{117}}\,\, \hfill \\
\end{gathered} \right.\)
\(\left( 2 \right)\,\,{a^2} + ab + {b^2} = 39\)
\(\left\{ \begin{gathered}
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {\sqrt {39} \,\,;\,\,0} \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = \sqrt {39} \,\,\, \hfill \\
\,\,{\text{Take}}\,\,\,\left( {a;b} \right) = \left( {0\,\,;\,\,\sqrt {39} } \right)\,\,\,\,\, \Rightarrow \,\,\,\,? = - \sqrt {39} \,\,\, \ne \sqrt {39} \,\, \hfill \\
\end{gathered} \right.\)
\(\left( {1 + 2} \right)\,\,\,\left\{ \begin{gathered}
\,\,\left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right) = {a^3} - {b^3} = 117 \hfill \\
\,\,{a^2} + ab + {b^2} = 39 \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,39\left( {a - b} \right) = 117\,\,\,\,\, \Rightarrow \,\,? = a - b\,\,\,{\text{unique}}\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
fskilnik.
_________________
Fabio Skilnik ::
GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here:
https://gmath.net