Giải thích các bước giải :
`a)A=x^2+20x+9`
`<=>A=(x^2+2×x×10+100)-91`
`<=>A=(x+10)^2-91`
Vì `(x+10)^2 ≥ 0`
`=>(x+10)^2-91 ≥ -91`
`=>A_(min)=-91`
Xảy ra dấu `=` khi :
`(x+10)^2=0<=>x+10=0<=>x=-10`
Vậy `A_(min)=-91` khi `x=-10`
`b)B=4x^2+5x+7`
`<=>B=(2x)^2+2×2x×5/4+(5/4)^2-(25)/(16)+(112)/(16)`
`<=>B=(2x+5/4)^2+(87)/(16)`
Vì `(2x+5/4)^2 ≥ 0`
`=>(2x+5/4)^2+(87)/(16) ≥ (87)/(16)`
`=>B_(min)=(87)/(16)`
Xảy ra dấu `=` khi :
`(2x+5/4)^2=0<=>2x+5/4=0<=>2x=-5/4<=>x=-5/8`
Vậy `B_(min)=(87)/(16)` khi `x=-5/8`
`c)C=3x^2-4x+5`
`<=>C=3(x^2-(4x)/3+5/3)`
`<=>C=3[x^2-2×x×2/3+(2/3)^2-4/9+(15)/9]`
`<=>C=3[x^2-2×x×2/3+(2/3)^2]+3×(15-4)/9`
`<=>C=3(x-2/3)^2+(11)/3`
Vì `(x-2/3)^2 ≥ 0`
`=>3(x-2/3)^2 ≥ 0`
`=>3(x-2/3)^2+(11)/3 ≥ (11)/3`
`=>C_(min)=-(11)/3`
Xảy ra dấu `=` khi :
`3(x-2/3)^2=0<=>x-2/3=0<=>x=2/3`
Vậy `C_(min)=(11)/3` khi `x=2/3`
`d)D=x^2-4xy+5y^2+10x-22y+28`
`<=>D=(x^2-4xy+4y^2)+(y^2-2y+1)+(10x-20y)+27`
`<=>D=(x-2y)^2+10(x-2y)+25+(y-1)^2+2`
`<=>D=(x-2y)^2+2×(x-2y)×5+5^2+(y-1)^2+2`
`<=>D=(x-2y+5)^2+(y-1)^2+2`
Vì `(x-2y+5)^2 ≥ 0; (y-1)^2 ≥ 0`
`=>(x-2y+5)^2+(y-1)^2+2 ≥ 2`
`=>D_(min)=2`
Xảy ra dấu `=` khi :
$\begin{cases}(x-2y+5)^2=0\\(y-1)^2=0\\\end{cases}$
`<=>`$\begin{cases}x-2y+5=0\\y-1=0\\\end{cases}$
`<=>`$\begin{cases}x-2×1+5=0\\y=1\\\end{cases}$
`<=>`$\begin{cases}x-3=0\\y=1\\\end{cases}$
`<=>`$\begin{cases}x=3\\y=1\\\end{cases}$
Vậy `D_(min)=2` khi `x=3; y=1`
