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 :: http://www.GMATH.net (Math for the GMAT)

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