Find roots for an equation with quadratic, linear and log terms?I'm wondering if there exists a closed-form or analytic expression for the roots of an equation of the formax2+bx+clogx=0.considering the natural log. Wolfram alpha is leading me to expressions involving the Lambert W (product log) function when I include either the quadratic term or the linear term (but not both) and analytic approximations when I supply real values for the coefficients.This is OK, but does a more general solution exist in terms of the coefficients?

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Find roots for an equation with quadratic, linear and log terms?I&#039;m wondering if there exists a closed-form or analytic expression for the roots of an equation of the formax2+bx+clogx=0.considering the natural log. Wolfram alpha is leading me to expressions involving the Lambert W (product log) function when I include either the quadratic term or the linear term (but not both) and analytic approximations when I supply real values for the coefficients.This is OK, but does a more general solution exist in terms of the coefficients?

Maffei2el · Accepted Answer

Step 1Any equation which can write  A  +  B  x  +  C  log  ⁡  (  D  +  E  x  )  =  0has a solution which can be expressed in terms of Lambert function.  x  =      C    B    W  (  F  )  −      D    E  where  F  =      B          C      E            e                            B          D          −          A          E                          C          E                    Step 2This is why Wolfram Alpha gave you solutions for  a      x    2    +  b  x  +  c  log  ⁡  x  =  0in the case where   a  =  0 or   b  =  0.However, for the general case you are considering   (  a  ≠  0  ,  b  ≠  0  ), I do not think that exists any analytical solution and only numerical methods (such as Newton) would solve the problem and this should be quite easy.

bucstar11n0h · Accepted Answer

Step 1The equation   a      x    2    +  b  x  +  c  log  ⁡  x  =  0 can be rewritten as (supposing a not null):      a    c    (  x  +      b          2      a            )    2    −            b      2              4      a      c        +  log  ⁡  (  x  )  =  0Taking the exponential of both members:      e          −              a        c            (      x      +              b                  2          a                            )        2            +                        b          2                          4          a          c                      =  xRemembering Lagrange series:  x  =      ∑          n      =      1                          D                  n          −          1                                                            e                          −              n                              a                c                            (              x              +                              b                                  2                  a                                                            )                2                            +                                                n                                      b                    2                                                                    4                  a                  c                                                              |                          x          =          0                            n      !        x  =      ∑          n      =      1            e          n                        b          2                          4          a          c                                        D                  n          −          1                                                            e                          −              n                              a                c                            (              x              +                              b                                  2                  a                                                            )                2                                              |                          x          =          0                            n      !      Step 2Supposing   a      /    c  &amp;gt;  0 and remembering Rodrigues formula of Hermite polynomials:      H    n    (  x  )  =  (  −  1      )    n        e                  x        2                        d      n              d              x        n                  e          −              x        2            we have:  x  =      ∑          n      =      1              ∞                                    (          −                                                    n                a                            c                                )                          n          −          1                            H                  n          −          1                            (                                            n              a                        c                                    b                      2            a                          )                    n      !

Find roots for an equation with quadratic, linear and log terms? I'm wondering if there exists a closed-form or analytic expression for the roots of an equation of the form ax^2+bx+c log x=0.

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