Solve { 5 a 2 − 4 a b − b 2 + 9 = 0 − 21 a 2 − 10 a b + 40 a − b 2 + 8 b − 12 = 0.

Question

Solve      {                            5                      a            2                    −          4          a          b          −                      b            2                    +          9          =          0                                      −          21                      a            2                    −          10          a          b          +          40          a          −                      b            2                    +          8          b          −          12          =          0.

Brendon Fernandez · Accepted Answer

You can notice that many terms of  (  b  +  5  a  −  4      )    2    =      b    2    +  10  a  b  −  8  b  −  40  a  +  25      a    2    +  16appear in the first equation. Similarly, in the first one, you can notice   (  b  +  2  a      )    2  .By algebraic manipulation you get that the original equations are equivalent to                    (        b        +        5        a        −        4                  )          2                                    =        4        (                  a          2                +        1        )                            (        b        +        2        a                  )          2                                    =        9        (                  a          2                +        1        )            which implies   4  (  b  +  2  a      )    2    =  9  (  b  +  5  a  −  4      )    2   and   2  (  b  +  2  a  )  =  ±  3  (  b  +  5  a  −  4  ). This should simplify things a little. (In each of the two possibilities you can express   b using   a as a linear expression. Then you will get a quadratic equation in   a. Or you can start by eliminating

migongoniwt · Accepted Answer

By arrange the terms, you can note that      {                            5                      a                      2                    −          4          a          b          −                      b            2                    +          9          =          0                                      −          21                      a                      2                    −          10          a          b          +          40          a          −                      b                      2                    +          8          b          −          12          =          0                          ⇔      {                            9          (                      a                      2                    +          1          )          =          (          2          a          +          b                      )                      2                                                4          (                      a                      2                    +          1          )          =          (          5          a          +          b          −          4                      )                      2                                  So we get      {                            9          (                      a                      2                    +          1          )          =          (          2          a          +          b                      )                      2                                                4          (          2          a          +          b                      )                      2                    =          9          (          5          a          +          b          −          4                      )                      2                                  According to the second equation, we can get two cases of first relation between   a and   b, then substitute them into the first equation respectively, we can get all the cases of values of pairs   (  a  ,  b  ).

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