Let $m$ and $n$ be the numbers of real roots of $x^2-12x+[x]+31=0$ and $x^2-5|x+2|-4=0$ respectively. Then $m^2+mn+n^2$ is equal to
Step-by-Step Solution
Key Concept: Equation 1: $x^2-12x+31=-[x]\geq-5$ needs $x\in[5,6)$ but LHS$=1>0$ at $x=5$; no solution, $m=0$. Equation 2: case analysis gives roots $-3,-2,7$; $n=3$.
$m=0,n=3$. $m^2+mn+n^2=9$.
Correct Answer: 9