A very simple question. How do we prove it? a x 2 + b x + c = a ( x − r 1 ) ( x − r 2 )

Question

A very simple question. How do we prove it?  a      x    2    +  b  x  +  c  =  a  (  x  −      r    1    )  (  x  −      r    2    )

Nathen Austin · Accepted Answer

For quadratic equation  0  =  a      x    2    +  b  x  +  c  =  a            (      x      +              b                  2          a                    )        2    −                    b        2            −      4      a      c              4              a        2              ,its roots are      r          1      ,      2        =            −      b      ±                        b          2                −        4        a        c                    2      a        .Depending on the sign of       b    2    −  4  a  c, the roots could be distinct reals, a multiple real, or distinct complexes. We actually do not need the fundamental theorem of algebra here, which is usually used for equations of higher orders (e.g., order 5 or more).Then we can evaluate it directly                    a        (        x        −                  r          1                )        (        x        −                  r          2                )                            =        a        [                  x          2                −        (                  r          1                +                  r          2                )        x        +                  r          1                          r          2                ]                                          =        a                  [                      x            2                    −                      (            −                          b              a                        )                    x          +                                                    b                2                            −              (                              b                2                            −              4              a              c              )                                      4                              a                2                                              ]                                                  =        a                  (                      x            2                    +                      b            a                    x          +                      c            a                    )                =        a                  x          2                +        b        x        +        c        .            I guess people are overthinking about this question.