SOLUTION: Find an equation with roots that are the negative reciprocals of the roots of ax^2 +bx +c =0

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Question 285939: Find an equation with roots that are the negative reciprocals of the roots of ax^2 +bx +c =0
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
ax%5E2+%2B+bx+%2B+c+=+0
x%5E2+%2B+%28b%2Fa%29%2Ax+%2B+%28c%2Fa%29+=+0
If the roots are r%5B1%5D and r%5B2%5D
%28x+-+r%5B1%5D%29%2A%28x+-+r%5B2%5D%29+=+0
x%5E2+-+%28r%5B1%5D+%2B+r%5B2%5D%29%2Ax+%2B+r%5B1%5D%2Ar%5B2%5D+=+0
Comparing the equations:
(1) -%28r%5B1%5D+%2B+r%5B2%5D%29+=+b%2Fa
(2) r%5B1%5D%2Ar%5B2%5D+=+c%2Fa
I'll say
r%5B01%5D+=+-%281%2Fr%5B1%5D%29
r%5B02%5D+=+-%281%2Fr%5B2%5D%29
and
r%5B1%5D+=+-%281%2Fr%5B01%5D%29
r%5B2%5D+=+-%281%2Fr%5B02%5D%29
----------------------
from (2)
%28-1%2Fr%5B01%5D%29%2A%28-1%2Fr%5B02%5D%29+=+c%2Fa
1%2F%28r%5B01%5D%2Ar%5B02%5D%29+=+c%2Fa
r%5B01%5D%2Ar%5B02%5D+=+a%2Fc
from (1)
-%28-1%2Fr%5B01%5D+-+1%2Fr%5B02%5D%29+=+b%2Fa
1%2Fr%5B01%5D+%2B+1%2Fr%5B02%5D+=+b%2Fa
multiply both sides by r%5B01%5D%2Ar%5B02%5D
r%5B02%5D+%2B+r%5B01%5D+=+%28b%2Fa%29%2Ar%5B01%5D%2Ar%5B02%5D
r%5B02%5D+%2B+r%5B01%5D+=+%28b%2Fa%29%2A%28a%2Fc%29
r%5B01%5D+%2B+r%5B02%5D+=+b%2Fc
and
x%5E2+-+%28r%5B01%5D+%2B+r%5B02%5D%29%2Ax+%2B+r%5B01%5D%2Ar%5B02%5D+=+0
x%5E2+-+%28b%2Fc%29%2Ax+%2B+a%2Fc+=+0
cx%5E2+-+bx+%2B+a+=+0
There could be a flaw somewhere -look it over