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Algebra Difficulty 4.6 AIME Find the answer

3. When x=13−2x=\frac{1}{\sqrt{3}-2}, the value of the function f(x)=x3+4x2−f(x)=x^{3}+4 x^{2}- 2x−62 x-6 is:

Pick one

Solution

3. B.

Notice that x=13−2=−(2+3)x=\frac{1}{\sqrt{3}-2}=-(2+\sqrt{3}), i.e., x+2=−3x+2=-\sqrt{3}.
Therefore, when x=13−2x=\frac{1}{\sqrt{3}-2},
f(x)=x(x+2)2−6(x+1)=3x−6(x+1)=−3(x+2)=33. \begin{array}{l} f(x)=x(x+2)^{2}-6(x+1) \\ \quad=3 x-6(x+1)=-3(x+2)=3 \sqrt{3} . \end{array}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.