2025 AMC 10A Problems/Problem 9: Difference between revisions
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==Solution== | ==Solution== | ||
The problem boils down to how many real roots does the equation <cmath>100(1-a)^3 - 300(1-a)^2 + 200(1-a) = 25</cmath> have? | The problem boils down to how many real roots does the equation <cmath>100(1-a)^3 - 300(1-a)^2 + 200(1-a) = 25</cmath> have? Using Descarte's Rule of Signs we find there shouldn't be any imaginary roots, so the answer is <imath>\boxed{3}</imath> real roots. | ||
~[[User:grogg007|grogg007]] | ~[[User:grogg007|grogg007]] | ||
Revision as of 13:44, 6 November 2025
Let 
. For how many real numbers 
does the graph of 
pass through the point 
?
(A) 
(B) 
(C) 
(D) 
(E) more than
Solution
The problem boils down to how many real roots does the equation ![\[100(1-a)^3 - 300(1-a)^2 + 200(1-a) = 25\]](http://latex-new.aopstest.com/f/f/8/ff8338b0bd12c4ae1c49d59ea8496bc430e48d8d.png)
have? Using Descarte's Rule of Signs we find there shouldn't be any imaginary roots, so the answer is 
real roots.
