\(\left(x+2\right)\left(x-2\right)-x\left(x-3\right)\)

\(=x^2-4-x^2+3x=3x-4\)

\(\left(X^2+2x+1\right)+\left(4y^2+\frac{4.1y}{4}+\frac{1}{16}\right)+2-\frac{1}{16}.\)

\(\left(x+1\right)^2+\left(2y+\frac{1}{4}\right)^2+\frac{15}{16}\ge\frac{15}{16}\)

\(x^2+4y^2+2x-y+2\)

\(=\left(x^2+2x+1\right)+\left[\left(2y\right)^2-2.2y.\frac{1}{4}+\left(\frac{1}{4}\right)^2\right]+\frac{15}{16}\)

\(=\left(x+1\right)^2+\left(2y-\frac{1}{4}\right)+\frac{15}{16}\)

Ta có: \(\hept{\begin{cases}\left(x+1\right)^2\ge0\forall x\\\left(2y-\frac{1}{4}\right)\ge0\forall y\end{cases}\Rightarrow\left(x+1\right)^2+\left(2y-\frac{1}{4}\right)+\frac{15}{16}\ge\frac{15}{16}}\)

Dấu " = " xảy ra \(\Leftrightarrow\hept{\begin{cases}\left(x+1\right)^2=0\\\left(2y-\frac{1}{4}\right)=0\end{cases}\Leftrightarrow\hept{\begin{cases}x+1=0\\2y-\frac{1}{4}=0\end{cases}\Leftrightarrow}\hept{\begin{cases}x=-1\\y=\frac{1}{8}\end{cases}}}\)

Vậy GTNN của \(x^2+4y^2+2x-y+2=\frac{15}{16}\Leftrightarrow\hept{\begin{cases}x=-1\\y=\frac{1}{8}\end{cases}}\)

Tham khảo nhé~

nhầm , chỗ kia phải là -y  , -4y/4

xy+x-2y=5

<=>xy-2y+x=5

<=>y(x-2)+x-2=5-2

<=>y(x-2)+(x-2)=3

<=>(y-1)(x-2)=3

tới đây tự làm tiếp

Ta có: x.y+x-2y = 5

=> x.(y+1)-2y = 5

=> x.(y+1)-2y-2 = 3

=> x.(y+1)-2.(y+1) = 3 

=> (x-2).(y+1) = 3

Lập bảng tìm x;y

Vậy x; y là: (-1;-2);(1;-4);(3;2);(5;0)

a) y x 5 = 1948 + 247

y x 5 = 2195

y = 2195 : 5

y = 439

b) y : 3 = 190 - 90

y : 3 = 100

y = 100 x 3

y = 300

c) y - 8357 = 3829 x 2

y - 8357 = 7658

y = 7658 + 8357

y = 16 015

d) y x 8 = 182 x 4

y x 8 = 728

y = 728 : 8

y = 91

a) y x 5 = 1948 + 247

y x 5 = 2195

y = 2195 :5

y = 439

b) y : 3 = 190 - 90

y : 3 = 100

y = 100 x 3

y = 300

c) y - 8357 = 3829 x 2

y - 8357 = 7658

y = 7658 + 8357

y = 16015

d) y x 8 = 182 x 4

y x 8 = 728

y = 728 : 8

y = 91

a) y x 5 = 1948 + 247

y x 5 = 2195

y = 2195 : 5

y = 439

b) y : 3 = 190 - 90

y : 3 = 100

y = 100 x 3

y = 300

c) y - 8357 = 3829 x 2

y - 8357 = 7658

y = 7658 + 8357

y = 16 015

d) y x 8 = 182 x 4

y x 8 = 728

y = 728 : 8

y = 91

Ta có : 

\(A=\frac{x^2+x+1}{\left(x+1\right)^2}\)

\(A=\frac{x^2+2x+1-x-1+1}{x^2+2x+1}\)

\(A=\frac{x^2+2x+1}{\left(x+1\right)^2}+\frac{-x-1}{\left(x+1\right)^2}+\frac{1}{\left(x+1\right)^2}\)

\(A=\frac{\left(x+1\right)^2}{\left(x+1\right)^2}-\frac{x+1}{\left(x+1\right)^2}+\frac{1^2}{\left(x+1\right)^2}\)

\(A=1-\frac{1}{x+1}+\left(\frac{1}{x+1}\right)^2\)

Đặt \(a=\frac{1}{x+1}\) ta có : 

\(A=1-a+a^2\)

\(A=a^2-a+1\)

\(A=\left(a^2-a+\frac{1}{4}\right)+\frac{3}{4}\)

\(A=\left(a-\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\)

Dấu "=" xảy ra khi và chỉ khi \(\left(a-\frac{1}{2}\right)^2=0\)

\(\Leftrightarrow\)\(a-\frac{1}{2}=0\)

\(\Leftrightarrow\)\(a=\frac{1}{2}\)

Do đó : 

\(a=\frac{1}{x+1}\)

\(\Leftrightarrow\)\(\frac{1}{2}=\frac{1}{x+1}\)

\(\Leftrightarrow\)\(x+1=2\)

\(\Leftrightarrow\)\(x=1\)

Vậy GTNN  của \(A\) là \(\frac{3}{4}\) khi \(x=1\)

Chúc bạn học tốt ~ 

1) 43 . 78 - 43 . 48 + 30 . 80 - 30 . 23

= 43.(78 - 48) + 30.(80 - 23)

= 43.30 + 30.57

= 30.(43 + 57)

= 30.100

= 3000

2) 31.175 - 31.50 + 69.125

= 31.(175 - 50) + 69.125

= 31.125 + 69.125

= 125.(31 + 69)

= 125.100

= 12500

3) 2.[(7 - 3¹³ : 3¹²) : 2² + 99] - 10²

= 2.[(7 - 3) : 4 + 99] - 100

= 2.(4 : 4 + 99) - 100

= 2.(1 + 99) - 100

= 2.100 - 100

= 200 - 100

= 100

4) 2²⁰¹⁹.2² : 2²⁰¹⁶ - 125 : 5² + 2019⁰

= 2²⁰²¹ : 2²⁰¹⁶ - 125 : 25 + 1

= 2⁵ - 5 + 1

= 32 - 4

= 28

\(2x^2+y^2+2x-2xy+5-4y=0\)

\(\Leftrightarrow\left[y^2-2y\left(x+2\right)+\left(x+2\right)^2\right]+\left(x^2-2x+1\right)=0\)

\(\Leftrightarrow\left(y-x-2\right)^2+\left(x-1\right)^2=0\)

\(\Leftrightarrow\left[{}\begin{matrix}y-x-2=0\\x-1=0\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}x=1\\y=3\end{matrix}\right.\)

\(S=\left(x+2\right)^2+\left(y-1\right)^2=\left(1+2\right)^2+\left(3-1\right)^2\)

\(=3^2+2^2=13\)

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