Find minimum without calculus

We can use calculus to find the minima, which happens when $x = ln(3)$ , then the minimum value we get is $9-18+1 = -8$ . However, it is asked that we should not use calculus, so take $e^x = p$ .

Then, we get a quadratic polynomial $g(x) = p^2 - 6p +1$ , and the minimum value of a quadratic equation occurs when $x = \frac {-b}{2a}$ ( we can derive that by the method of completing the square) and

the minimum value is $\frac {4ac - b^2}{4a}$ , here it is nothing but

$\frac {4-36}{4}$ = -8

Completing squares we get the following expression: $f(x)=(e^{x}-3)^2-8$ . Then $f(x) \geq -8$ so 8 is an lower bound therefore a candidate to minimun value and we reach the minimun value when $e^x-3=0$ . But this is happens when $x = \ln(3)$