a,
$\Delta'= (m-1)^2-m+5$
$= m^2-2m+2-m+5$
$= m^2-3m+7$
$= m^2-2m.1,5+2,25+4,75$
$= (m-1,5)^2+4,75>0$ (luôn đúng)
=> đpcm
b,
Theo Viet:
$x_1+x_2=2(m-1)$
$x_1x_2=m-5$
$\Rightarrow x_1^2+x_2^2= (x_1+x_2)^2-2x_1x_2= (2m-2)^2-2(m-5)= 4m^2-8m+4-2m+10= 4m^2-10m+14= (2m)^2-2.2m.2,5+6,25+7,75= (2m-2,5)^2+7,75\ge 7,75$
$min=7,75\Leftrightarrow m=1,25$
c,
Hệ thức liên hệ:
$x_1+x_2-2x_1x_2= 2m-2-2m+10= 8$