SOLUTION: if the sum of the roots of quadratic equation, ax^2+bx+c=0, is equal to the sum of cubes of their reciprocals then show that ab^2=3a^2c+c^3

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Question 850743: if the sum of the roots of quadratic equation, ax^2+bx+c=0, is equal to the sum of cubes of their reciprocals then show that ab^2=3a^2c+c^3
Found 2 solutions by Edwin McCravy, AnlytcPhil:
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
I had a couple of typos on the first solution I submitted.
Below I have corrected them.

Edwin
Answer by AnlytcPhil(1806)   (Show Source): You can put this solution on YOUR website!
Question 850743
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if the sum of the roots of quadratic equation, ax^2+bx+c=0, is equal to the sum of cubes of their reciprocals then show that ab^2=3a^2c+c^3
The sum of the roots of  is 

Use the quadratic formula to find the sum of the cubes of the reciprocals of the roots:



To make things easier, let 



Combine fractions on the left



Divide through by 





The numerator factors as the sum of two cubes:

 




The denominator simplifies:





So we have



Since 


Substitute for z²





 





Simplify the fraction on the left 



Divide both sides by -b



Multiply both sides by 8a³



Cross multiply:





Edwin
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