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As easy as abc (Posted on 2011-06-28) Difficulty: 3 of 5
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?

No Solution Yet Submitted by Ady TZIDON    
Rating: 5.0000 (1 votes)

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Some Thoughts My Attempt | Comment 5 of 10 |

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
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