Difference between revisions of "AoPS Wiki talk:Problem of the Day/August 2, 2011"
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<cmath>(x_2,y_2) = (\sqrt[3]{218 + \psi^3},\psi)</cmath> | <cmath>(x_2,y_2) = (\sqrt[3]{218 + \psi^3},\psi)</cmath> | ||
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+ | There are more solutions in <math>\mathbb{C}^2</math>, corresponding to the complex roots of <math>\frac{6740}{72}+\frac{1348}{72}y-\frac{19}{18}y^2+y^3</math>. |
Latest revision as of 04:04, 2 August 2011
Problem: Find all such that and .
Solution: Let and . Then and . Hence, This quartic has two real roots; and by the cubic formula. Call this second root (which evaluates to about -3.01) . The two real solutions are therefore: and
There are more solutions in , corresponding to the complex roots of .