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Problem 705

AIME late

Algebra Difficulty 5.2 Find the answer

I3.1 Given that $\frac{1-\sqrt{3}}{2}$ satisfies the equation $x^{2}+p x+q=0$ , where $p$ and $q$ are rational numbers. If $A=|p|+2|q|$ , find the value of $A$ .

Your answer

Official solution

For an equation with rational coefficients, conjugate roots occur in pairs. That is, the other root is $\frac{1+\sqrt{3}}{2}$ .
$\begin{array}{l} \frac{1-\sqrt{3}}{2}+\frac{1+\sqrt{3}}{2}=-p \Rightarrow p=-1 \\ \frac{1-\sqrt{3}}{2} \times \frac{1+\sqrt{3}}{2}=q \Rightarrow q=-\frac{1}{2} \\ A=|-1|+2\left|-\frac{1}{2}\right|=2 \end{array}$

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