Difference between revisions of "AoPS Wiki talk:Problem of the Day/August 2, 2011"
BoyosircuWem (talk | contribs) (Created page with "Let <math>X=x^3</math> and <math>Y=y^3</math>. Then <math>X=218+Y</math> and <math>(218+Y)^{2/3} = 24 + Y^{2/3}</math>. Hence, <cmath>218+y^3=(24+y^2)^3</cmath> <cmath>(y-5)(674...") |
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− | Let <math>X=x^3</math> and <math>Y=y^3</math>. Then <math>X=218+Y</math> and <math>(218+Y)^{2/3} = 24 + Y^{2/3}</math>. Hence, | + | ''Problem'': Find all <math>(x,y)\in\mathbb{R}^2</math> such that <math>x^2-y^2=24</math> and <math>x^3-y^3=218</math>. |
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+ | ''Solution'': Let <math>X=x^3</math> and <math>Y=y^3</math>. Then <math>X=218+Y</math> and <math>(218+Y)^{2/3} = 24 + Y^{2/3}</math>. Hence, | ||
<cmath>218+y^3=(24+y^2)^3</cmath> | <cmath>218+y^3=(24+y^2)^3</cmath> | ||
<cmath>(y-5)(6740+1348y-76y^2+72y^3)=0</cmath> | <cmath>(y-5)(6740+1348y-76y^2+72y^3)=0</cmath> | ||
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and | and | ||
<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 .