$\displaystyle f(x) = \sqrt{x^2-5x+6}$

$\displaystyle f(x) = \sqrt{(x-2)(x-3)}$

$\displaystyle (x-2)(x-3) \geq 0$

$\displaystyle x-2 \geq 0 \Longrightarrow x \geq 2$

$\displaystyle x-3 \geq 0 \Longrightarrow x \geq 3$

I know from the graph that it should be $\displaystyle x \leq 2$

So what am I doing wrong?