If x - y = 3 then x 3 - y 3 = ?

Question

If       x    -    y    =    3   then             x      3        -          y      3        =   ?

Dalton Lester · Accepted Answer

Step 1We know that in general                              XXX                            (              x        3            -              y        3            )        =          (      x      -      y      )              (              x        2            +      x      y      +              y        2            )      and since                              XXX                      x    -    y    =    3  this gives us                              XXX                            (              x        3            -              y        3            )        =    3          (              x        2            +      x      y      +              y        2            )      There is no single solution.The table below shows some of the possible combinations:                    Given                 x        −        y        =        3                          In which case                            if                 x        =                    then                 y        =                              x                      3                          −                  y                      3                          =                            −        3                    −        6                    189                            −        2                    −        5                    117                            −        1                    −        4                    63                            0                    −        3                    27                            1                    −        2                    9                            2                    −        1                    9                            3                    0                    27                            4                    1                    63                            5                    2                    117                            6                    3                    189

Janessa Olson · Accepted Answer

Step 1From equation (1)       y    =    x    -    3  Substitution in (2) gives3)             x      3        -                  (        x        -        3        )            3        =    ?  Consider just the                     (        x        -        3        )            3                  (      x      -      3      )              (      x      -      3      )              (      x      -      3      )                  (      x      -      3      )              (              x        2            -      6      x      +      9      )                  x      3        -    9          x      2        +    27    x    -    27  Substituting back into (3)            x      3        -          (              x        3            -      9              x        2            +      27      x      -      27      )            9          x      2        -    27    x    +    27  Set this to equal y      ⇒    y    =    9          x      2        -    27    x    +    27  Factor out the 94)       y    =    9          (              x        2            -      3      x      +      3      )      Step 2From this point on it depends what you wish to do with itDetermine the vertexConsider the       -    3    x   in equation (4)Apply             (      -              1        2            )        ×    -    3    =    +          3      2                                    x                                    vertex                                      =        +                  3          2                    By substitution in (4)                              y                                    vertex                                      =        9                  [                                    (                              3                2                            )                        2                    -          3                      (                          3              2                        )                    +          3          ]                =        9                  [                      3            4                    ]                =                  27          4                                                  Vertex                →                  (          x          ,          y          )                →                  (                      3            2                    ,                      27            4                    )

If <mstyle displaystyle="true"> x - y = 3 </mstyle> then <msty

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