If $a$ is a real root of the equation $x^5-x^3+x-2=0$ such that $[a^6]=3k$. Find $k$.
Here, $[x]$ denotes the Greatest Integer function.
My try:
First I plugged in $1$ in the equation. I got $-1$, which is close to $0$ so the root must be near 1. Then I plugged in $1.1$, I got $-0.62049$. The sign didn't change, so it must be greater than $1.1$. Then I plugged in $1.2$ and I got $-0.03968$ which is very close to $0$ but the sign still didn’t change. So, now I plugged in $1.3$ and I got $0.81593$. The sign did change, but it got farther away from $0$, so the root must be very close to $1.2$. So I assumed $a=1.21$, and with this I got $k=1$.
I was fortunate that this was the correct answer, but I highly doubt this is the correct way to solve this problem. Please tell me if there is any other efficient and correct way of doing this.