Question 1144876
<pre>{{{ax^2 + 2bx + 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

{{{(r)(1/r^2)=c/a}}}

{{{1/r=c/a}}}

{{{a=rc}}}

{{{a/c=r}}}

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

{{{a(a/c)^2 + 2b(a/c) + c = 0}}}

{{{a(a^2/c^2) + 2ab/c + c = 0}}}

{{{a^3/c^2+2ab/c+c=0}}}

{{{a^3+2abc+c^3=0}}}

{{{a^3+c^3+2abc=0}}}

Edwin</pre>