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chihiroasleaf

  • 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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  1. chihiroasleaf
    • one year ago
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    If we add both equations, we'll get \[x^3 - 3xy^2 + 3x^2y - y^3 = 52 \] While \[ (x-y)^3 = x^3 - 3x^2y + 3xy ^2 - y^3 \]

  2. mathmath333
    • one year ago
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    u have to substract both equations

  3. chihiroasleaf
    • one year ago
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    Umm What I mean above is If we subtract, we get \[ x^3 - 3xy^2 + 3x^2y - y^3 = 36\]

  4. ParthKohli
    • one year ago
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    \[(x+iy)^3 = x^3 + 3x^2y i - 3xy^2 - iy^3 = (x^3 - 3x y^2) + (3x^2 y - y^3) i \]\[(x-iy)^3 = (x^3 - 3x y^2) + (y^3 - 3x ^2 y) i\]\[x^2 + y^2 = (x+iy)(x-iy)\]

  5. ParthKohli
    • one year ago
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    \[\Rightarrow (x+iy)^3 = 44-8i, (x-iy)^3 = 44 + 8i\]\[\Rightarrow (x^2+y^2)^3 = (44+8i)(44-8i) = 2000\]\[\Rightarrow x^2 + y^2 = \sqrt[3]{2000}\]

  6. chihiroasleaf
    • one year ago
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    Thank you @ParthKohli :) How did you get the idea?

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

  8. chihiroasleaf
    • one year ago
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    I didn't think to use conplex numbers :-|

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