Let m and n be the numbers of real roots of the quadratic equations x^2-12x+[x]+31=0 and x…

Question

Let m and $n$ be the numbers of real roots of the quadratic equations ${x}^{2}-12x+[x]+31=0$ and ${x}^{2}-5|x+2|-4=0$ respectively, where $[x]$ denotes the greatest integer $\le x$. Then ${m}^{2}+mn+{n}^{2}$ is equal to \_\_\_\_\_\_\_\_\_\_.

Answer: 9

Accepted Answer

First equation: writing $\{x\}=x-[x]$, the equation gives $\{x\}=x^2-11x+31$. For a valid fractional part, $0\le x^2-11x+31<1$, which gives $x\in(5,6)$, so $[x]=5$.

Substituting $[x]=5$: $x^2-12x+36=0 \implies x=6$, but $6
otin(5,6)$ --- contradiction. So no solution: $m=0$.

Second equation: solving $x^2-5|x+2|-4=0$ piecewise gives three real roots $x=7,-2,-3$: $n=3$.

$m^2+mn+n^2=0+0+9=9.$

Solutions · 1