Track / Stage 5 / 137 of 400 #737 of 2000

Problem 737

AIME late

Algebra Difficulty 5.3 Find the answer

I4.2 Let $[x]$ be the largest integer not greater than $x$ . For example, $[2.5]=2$ .
If $b$ satisfies the system of equations $\left\{\begin{aligned} A x^{2}-4 & =0 \\ 3+2(x+[x]) & =0\end{aligned}\right.$ , find the value of $b$ .

Your answer

Official solution

$\left\{\begin{aligned} 16 x^{2}-4 & =0 \\ 3+2(x+[x]) & =0\end{aligned}\right.$ from the first equation $x=\frac{1}{2}$ or $-\frac{1}{2}$ .
Substitute $x=\frac{1}{2}$ into the second equation: LHS $=3+2\left(\frac{1}{2}+0\right)=4 \neq$ RHS
Substitute $x=-\frac{1}{2}$ into the second equation: LHS $=3+2\left(-\frac{1}{2}-1\right)=0=$ RHS
$\therefore b=-\frac{1}{2}$

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