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2025 AMC 10A Problems/Problem 9: Difference between revisions

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(A) <imath>1</imath> (B) <imath>2</imath> (C) <imath>3</imath> (D) <imath>4</imath> (E) more than <imath>4</imath>
(A) <imath>1</imath> (B) <imath>2</imath> (C) <imath>3</imath> (D) <imath>4</imath> (E) more than <imath>4</imath>
==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? We can divide by <imath>25</imath> and use Descarte's Rule of Signs to get <imath>\boxed{3}</imath> real roots.
~[[User:grogg007|grogg007]]

Revision as of 13:44, 6 November 2025

Let $f(x) = 100x^3 - 300x^2 + 200x$. For how many real numbers $a$ does the graph of $y = f(x - a)$ pass through the point $(1, 25)$?

(A) $1$ (B) $2$ (C) $3$ (D) $4$ (E) more than $4$

Solution

The problem boils down to how many real roots does the equation \[100(1-a)^3 - 300(1-a)^2 + 200(1-a) = 25\] have? We can divide by $25$ and use Descarte's Rule of Signs to get $\boxed{3}$ real roots. ~grogg007

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