How can I prove that for any real ( x , y ) root this system of equations x 4 + y 2 = ( a + 1 a ) 3 x 4 − y 2 = ( a − 1 a ) 3 the inequation x 2 + | y | ≥ 4 is always true?Also, when will x 2 + | y | = 4 be true for this system?

Question

How can I prove that for any real   (  x  ,  y  ) root this system of equations      x    4    +      y    2    =        (    a  +      1    a                  )        3        x    4    −      y    2    =        (    a  −      1    a                  )        3  the inequation       x    2    +  |  y  |  ≥  4 is always true?Also, when will       x    2    +  |  y  |  =  4 be true for this system?

Keegan Barry · Accepted Answer

Hint:       x    4    =            (              x        4            +              y        2            )      +      (              x        4            −              y        2            )        2   and       y    2    =            (              x        4            +              y        2            )      −      (              x        4            −              y        2            )        2  So       x    4    =      a    3    +  3  a  ∗      1          a      2        =      a    3    +  3      1    a   and       y    2    =  3      a    2    ∗      1    a    +            1      a        3    =  3  a  +      1          a      3       (and presumably   a  &amp;gt;  0 else we&#039;d have       1    0   or       x    4    &amp;lt;  0)So       x    2    +  |  y  |  =            a      3        +    3          1      a        +      3    a    +          1              a        3             so by AM-GM      x    2    +  |  y  |  ≥  2            (              a        3            +      3              1        a            )      (      3      a      +              1                  a          3                    )        4    =  2            3              a        4            +      9      +      1      +              3                  a          4                      4  And applying the AM-GM result that for   x  &amp;gt;  0 that  x  +      1    x    ≥  2      x          1      x        =  2 we have      x    2    +  |  y  |  ≥  2            3              a        4            +      9      +      1      +              3                  a          4                      4    =  2            10      +      3      (              a        4            +              1                  a          4                    )        4    ≥  2            10      +      3      ∗      2        4    =  2      16    4    =  2  ∗  2  =  4

racodelitusmn · Accepted Answer

Hint:Add and subtract the equations and divide by   2 to get      x    4    =      a    3    +            3      a       and       y    2    =            1              a        3              +  3  a

How can I prove that for any real ( x , y ) root this system of equations x

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New Questions in Algebra I