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anonymous
 one year ago
If x and y are real numbers, with
\[x^3  3xy^2 = 44\] and
\[ y^3  3x^2y = 8 \]
What is the value of \[ x^2 + y^2 \]
anonymous
 one year ago
If x and y are real numbers, with \[x^3  3xy^2 = 44\] and \[ y^3  3x^2y = 8 \] What is the value of \[ x^2 + y^2 \]

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anonymous
 one year ago
Best ResponseYou've already chosen the best response.0If we add both equations, we'll get \[x^3  3xy^2 + 3x^2y  y^3 = 52 \] While \[ (xy)^3 = x^3  3x^2y + 3xy ^2  y^3 \]

mathmath333
 one year ago
Best ResponseYou've already chosen the best response.0u have to substract both equations

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Umm What I mean above is If we subtract, we get \[ x^3  3xy^2 + 3x^2y  y^3 = 36\]

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.7\[(x+iy)^3 = x^3 + 3x^2y i  3xy^2  iy^3 = (x^3  3x y^2) + (3x^2 y  y^3) i \]\[(xiy)^3 = (x^3  3x y^2) + (y^3  3x ^2 y) i\]\[x^2 + y^2 = (x+iy)(xiy)\]

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.7\[\Rightarrow (x+iy)^3 = 448i, (xiy)^3 = 44 + 8i\]\[\Rightarrow (x^2+y^2)^3 = (44+8i)(448i) = 2000\]\[\Rightarrow x^2 + y^2 = \sqrt[3]{2000}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Thank you @ParthKohli :) How did you get the idea?

ParthKohli
 one year ago
Best ResponseYou've already chosen the best response.7I don't know either. Maybe the \(x^2 + y^2\) triggered me and I started thinking along the lines of complex numbers.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0I didn't think to use conplex numbers :
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