proof related to expected values

Question

I wanna prove following expression. $$ n\sum_{i=1}^{n} {X_i^2} - (\sum_{i=1}^{n}{X_i})^2 = n\sum_{i=1}^{n}{(X_i-\bar X)^2}$$

pf) $$ n\sum_{i=1}^{n} {X_i^2} - (\sum_{i=1}^{n}{X_i})^2 = n\sum_{i=1}^{n} {X_i^2} - (n \bar X)^2$$ $$ = n\sum_{i=1}^{n} {X_i^2} -n^2 \bar X^2 = n(\sum_{i=1}^{n} {X_i^2} - n \bar X^2)$$

I'm struggling from here. Thank you for your help.

Answer 1

$\Sigma_{i=1}^n(X_i-\bar{X})^2\\ =\Sigma_{i=1}^n(X^2_i-2X_i\bar{X}+\bar{X}^2)\\ = \Sigma_{i=1}^nX^2_i-2n\bar{X}^2+n\bar{X}^2\\=\Sigma_{i=1}^nX^2_i-n\bar{X}^2$

Now just multiply both sides with $n$