The Question
If the equation $sin^2x -k\space sinx -3 =0$ has exactly two distinct real roots in $[0,\pi]$, then find the values of $k$.
The Answer
Let $f(t)=t^2 -kt -3$, where $t=sinx$.
Since equation has exactly two distinct real roots in $[0,\pi]$, $f(t)=0$ must have exactly one root in $(0,1)$.
Now, $f(0)=-3$.
So, we must have $f(1)=-k-2>0$
or $k<-2$
My Question
Why does $f(t)=0$ have exactly one root in $(0,1)$ ? Why not $2$ ? I guess this might be a trivial question but I really don't get it.