Ta có

x^2+5y^2+2x-4xy-10y+14 
=[x^2+2x(1-2y)+(1-2y)^2]+y^2-6y+13 
=(x+1-2y)^2+(y^2-2y.3+9)+4 
=(x+1-2y)^2+(y-3)^2+4. 
mà

(x+1-2y)^2 > hoặc=0 với mọi x,y thuộc R 
và (y-3)^2 > hoặc=0 với mọi y thuộc R 
=> (x+1-2y)^2+(y-3)^2+4 > hoặc =4 với mọi x,y thuộc R 
=> (x+1-2y)^2+(y-3)^2+4 >0 với mọi x,y thuộc R.

Akai Shuichi:chép sai đề rồi con ơi :D

a/ \(x^2+xy+y^2+1\)=\(\left(x^2+2x\dfrac{y}{2}+\left(\dfrac{y}{2}\right)^2\right)+\dfrac{3y^2}{4}+1\)

=\(\left(x+\dfrac{y}{2}\right)^2+\dfrac{3y^2}{4}+1\) \(\ge\)0

vậy....

b

Câu hỏi của KiKyo - Toán lớp 8 - Học toán với OnlineMath

Em tham khảo nhé!

Đặt \(A=x^2+5y^2+2x-4xy-10y+14\)

\(A=\left(x^2-4xy+4y^2\right)+\left(2x-4y\right)+1+y^2-6y+9+4\)

\(A=\left(x-2y\right)^2+2\left(x-2y\right)+1+\left(y-3\right)^2+4\)

\(A=\left(x-2y+1\right)^2+\left(y-3\right)^2+4\ge4>0\)

\(\Rightarrow A>0\left(đpcm\right)\)

a/ \(x^2+xy+y^2+1=\left(x^2+xy+\frac{y^2}{4}\right)+\frac{3y^2}{4}+1=\left(x+\frac{y}{2}\right)^2+\frac{3y^2}{4}+1>0\)

b/ \(x^2+5y^2+2x-4xy-10y+14\)

\(=\left(x^2-4xy+4y^2\right)+2\left(x-2y\right)+1+\left(y^2-6y+9\right)+4\)

\(=\left(x-2y\right)^2+2\left(x-2y\right)+1+\left(y-3\right)^2+4\)

\(=\left(x-2y+1\right)^2+\left(y-3\right)^2+4>0\)

Giải:

a) \(x^2+xy+y^2+1\)

\(=x^2+2.x.\dfrac{y}{2}+\left(\dfrac{y}{2}\right)^2+\dfrac{3y^2}{4}+1\)

\(=\left(x^2+2.x.\dfrac{y}{2}+\left(\dfrac{y}{2}\right)^2\right)+\dfrac{3y^2}{4}+1\)

\(=\left(x+\dfrac{y}{2}\right)^2+\dfrac{3y^2}{4}+1\ge1>0;\forall x\)

Vậy ...

Ta có : \(x^2+5y^2+2x-4xy-10y+14\)

\(=x^2+2x\left(1-2y\right)+\left(1-2y\right)^2-\left(1-2y\right)^2+5y^2-10y+14\)

\(=\left(x-2x+1\right)^2-1-4y^2+4y+5y^2-10y+14\)

\(=\left(x-2x+1\right)^2+y^2-6y+9+4\)

\(=\left(x-2x+1\right)^2+\left(y-3\right)^2+4\ge4>0\) (đpcm)

Ta có: x² + 5y² + 2x - 4xy - 10y + 14 

= (x² - 4xy + 4y²) + (2x - 4y) + 1 + (y² - 6y + 9) + 4

= (x - 2y)² + 2(x - 2y) + 1 + (y - 3)² + 4

= (x - 2y + 1)² + (y - 3)² + 4 > 0 \(\forall\)x; y

Do (x - 2y + 1)² \(\ge\)0; (y - 3)² \(\ge\)0 ; 4 > 0

\(x^2+5y^2+2x-4xy+10y+14\)

\(=\left[x^2+2x\left(1-2y\right)+\left(1-2y\right)^2\right]+y^2-6y+13\)

\(=\left(x+1-2y\right)^2+\left(y^2-2y\cdot3+9\right)+4\)

\(=\left(x+1-2y\right)^2+\left(y-3\right)^2+4\)

Ta có: \(\hept{\begin{cases}\left(x+1-2y\right)^2\ge0\forall x\inℝ\\\left(y-3\right)^2\ge0\forall x\inℝ\end{cases}}\)

=> \(\left(x+1+2y\right)^2+\left(y-3\right)^2+4\ge4\)

=> \(x^2+5y^2+2x-4xy-10y+14>0\)(đpcm)

       \(x^2+4y^2+z^2-2x-6z+8y+15\)

\(=\left(x^2-2x+1\right)+\left(4y^2+8y+4\right)+\left(z^2-6z+9\right)+1\)

\(=\left(x-1\right)^2+4\left(y+1\right)^2+\left(z-3\right)^2+1>0\forall x;y\)

       \(x^2+5y^2+2x-4xy-10y+14\)

\(=\left(x^2-4xy+4y^2\right)+\left(2x-4y\right)+1+y^2-6y+9+4\)

\(=\left(x-2y\right)^2+2\left(x-2y\right)+1+\left(y-3\right)^2+4\)

\(=\left(x-2y+1\right)^2+\left(y-3\right)^2+4>0\forall x;y\)

Chúc bạn học tốt.