`color{#bfbfbf}{#}color{#858585}{d}color{#424242}{t}color{#050505}{t}`

`a)`

`2x^2-(m+3)x+m=0`

`\Delta=[-(m+3)]^2-4*2m=m^2-2m+9=m^2-2m+1+8=(m-1)^2+8>=8AAm\inRR`

`->` Pt có `2` nghiệm pb với mọi `m`

`b)`

`{(x_1+x_2=(m+3)/2),(x_1x_2=m/2):}` (Viète)

Theo đề:

`P=|x_1-x_2|`

`P=\sqrt((x_1+x_2)^2-4x_1x_2)`

`P=\sqrt(((m+3)/2)^2-4*m/2)`

`P=\sqrt((m^2+6m+9)/4-2m)`

`P=\sqrt((m^2-2m+9)/4)`

`P=(\sqrt(m^2-2m+1+8))/2`

`P=(\sqrt((m-1)^2+8))/2`

Có `(m-1)^2>=0AAm\inRR`

`-> (m-1)^2+8>=8`

`-> \sqrt((m-1)^2+8)>=\sqrt(8)`

`-> P=(\sqrt((m-1)^2+8))/2>=(\sqrt(8))/2=\sqrt(2)`

Dấu "`=`" xảy ra khi `P=\sqrt(2)->m=1`

Vậy `minP=\sqrt(2)` khi `m=1`