SOLUTION: One root of the equation ax^2 + 2bx + c = 0 is the reciprocal of the square of the other root. Show that a^3 +c^3 +2abc=0.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: One root of the equation ax^2 + 2bx + c = 0 is the reciprocal of the square of the other root. Show that a^3 +c^3 +2abc=0.      Log On


   

Question 1144876: One root of the equation ax^2 + 2bx + c = 0 is the reciprocal of the square of the other root. Show that a^3 +c^3 +2abc=0.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
ax%5E2+%2B+2bx+%2B+c+=+0 

Let r be one root.  Then the other root is 1/r²

We know that the constant term of a quadratic divided by the leading
coefficient equals the product of the roots.

So the product of the two roots is c/a, therefore

%28r%29%281%2Fr%5E2%29=c%2Fa

1%2Fr=c%2Fa

a=rc

a%2Fc=r

So a/c is a root and must satisfy the original equation:

a%28a%2Fc%29%5E2+%2B+2b%28a%2Fc%29+%2B+c+=+0

a%28a%5E2%2Fc%5E2%29+%2B+2ab%2Fc+%2B+c+=+0

a%5E3%2Fc%5E2%2B2ab%2Fc%2Bc=0

a%5E3%2B2abc%2Bc%5E3=0

a%5E3%2Bc%5E3%2B2abc=0

Edwin