Why does quadratic utility function imply $\mu-\sigma$ preference? Why does investors having quadratic utility function mean that their optimal portfolios can be chosen by only considering mean and variance of returns i.e. imply $\mu-\sigma$ preference?
If you have quadratic preferences then your utility function is:
$$ U(W) =  W - \lambda W^2$$
this implies your expected utility function looks like:
$$ E[U(W)] =  E[W - \lambda W^2] =  E[W] - \lambda E[W^2]$$
$$ =  E[W] - \lambda E[W^2 - E[W]^2 + E[W]^2]$$
$$ =  E[W] - \lambda E[W^2 - E[W]^2] + \lambda E[E[W]^2]]$$
$$ =  \mu_w - \lambda \sigma_w^2 + \lambda \mu_w^2$$

Therefore, we have established that expected utility depends only only the mean $(\mu_W)$ and variance $(\sigma^2_W)$ of wealth and the risk parameter $(\lambda)$.