Question

Let $f\left(x\right)=a{x}^2+bx+c$, where $a\ne 0,a,b,c$ are integers and $f\left(1\right)=1$, $6<f\left(3\right)<8$ and $18<f\left(5\right)<22$, then the number of solutions of the equation $f\left(x\right)=e^x$ is

Answer 1

$a+b+c=\mathrm{1.........}\left(1\right)9a+3b+1=\mathrm{7........}\left(2\right)18<25a+5b+1<\mathrm{22.......}\left(3\right)$

From the above equaitons,
$\Rightarrow 4<7a-b-c<8\Rightarrow 4<7a-b+a+b-1<8\Rightarrow 5<8a<9\Rightarrow a=1,b=-1,c=1$

$f\left(x\right)=x^2-x+1$