The required values of x are x = {13, 14}.

Given

$\rm x^{2} - 27 x + 182 = 0$

$\rm or, x^{2} - (13 + 14) x + 182 = 0$

$\rm or, x^{2} - 13x - 14x + 182 = 0$

$\rm or, x (x - 13) - 14( x - 13) = 0$

$\rm or, (x - 14)(x - 13) = 0$

Either

$\rm (x - 14) = 0$

$\rm \therefore x = 14$

Or

$\rm (x - 13) = 0$

$\rm \therefore x = 13$