Given an equation ax^2+bx+c=0 what values a,b cause the roots of the equation to be a and b and the discriminant to be equal to c?

Here is my attempt.

x=a ==> a*a^2 + b*a + c = 0   (1)

x=b ==> a*b^2 + b*b + c = 0   (2)

Subtracting (2) from (1) gives

a^3 - a*b^2 + b*a - b^2 = 0

       or

(a-b)*(a^2 + b*a + b) = 0

   a^2 + b*a + b = 0  gives

           -a^2
      b = ------
            a+1

      b^2 - 4*a*c = c  gives

         a^4 - 4*c*a^3 - 9*c*a^2 - 6*c*a - c = 0

         a = ??? in terms of c

   a-b = 0  gives

      b = a

      b^2 - 4*a*c = c  gives

         a^2 - 4*c*a - c = 0

         Therefore,

            b = a = 2*c +- sqrt(4*c^2 + c)

Posted by Bractals on 2011-06-30 14:45:27