2025 AMC 12A Problems/Problem 17
The polynomial 
has three roots in the complex plane, where 
. What is the area of the triangle formed by these three roots?
Solution 1 (Symmetry)
Let 
be a complex number such that 
then we can change our polynomial to the following,
![\[(z+i)(z+2i)(z+3i) + 10 = (c-i)(c)(c+i)+10 = c^3 +c+10 = 0.\]](http://latex-new.aopstest.com/b/0/0/b00ec5ed4b90ce45d91e37a9cd7890bc4346aba8.png)
Notice 
is a root, and from this we get a quadratic and find the the possible values of 
Now we simply subtract 
from each root to get the roots to be 
and 
Moving back to the coordinate plane our points are 
and 
and using shoelace gives us an area of 
Note
We don't have to shift the coordinates back since area stays same regardless of position. Therefore, we can use 
and 
, which has an area of 
, the same result.
Solution 2 (bash)
Expand the left hand side, and we get 
. We immediately see that 
is a root, so factor this out, and we get 
. We put this into the quadratic formula, and we get the other two roots are 
. Note that 
, hence we get 
and 
are the other two roots. We convert into coordinates to get 
, 
, and 
. Note that one of these lines is vertical ( to ), so the area is the base (
) times the height (
) over 
, aka 
.
~ScoutViolet
See Also
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