a) \(x^2+xy+y^2+1\)

\(=x^2+xy+\dfrac{y^2}{4}-\dfrac{y^2}{4}+y^2+1\)

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

mà \(\left\{{}\begin{matrix}\left(x+\dfrac{y}{2}\right)^2\ge0,\forall x;y\\\dfrac{3y^2}{4}\ge0,\forall x;y\end{matrix}\right.\)

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

\(\Rightarrow dpcm\)

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

\(=\left(x-1\right)^2+4\left(y^{ }+1\right)^2+\left(z-3\right)^2+1>0,\forall x.y\)

\(\Rightarrow dpcm\)

Ta có: \(x^2+4y^2+z^2-2x+8y+15\)

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

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

(vì \(\left(x-1\right)^2\ge0;\left(2y+2\right)^2\ge0;\left(z-3\right)^2\ge0\))

Vậy không có giá trị nào của x,y,z thỏa mãn đẳng thức đề bài cho

\(3x^2+6y^2-12x-20y+40=0\)

\(\Rightarrow\left(3x^2-12x+12\right)+\left(6y^2-12y+6\right)+22=0\)

\(\Rightarrow3\left(x^2-4x+4\right)+6\left(y^2-2y+1\right)+22=0\)

\(\Rightarrow3\left(x-2\right)^2+6\left(y-1\right)^2+22=0\)

Ta thấy: \(3\left(x-2\right)^2\ge0\forall x\)

              \(6\left(y-1\right)^2\ge0\forall y\)

\(\Rightarrow3\left(x-2\right)^2+6\left(y-1\right)^2\ge0\forall x;y\)

\(\Rightarrow3\left(x-2\right)^2+6\left(y-1\right)^2+22>0\forall x;y\)

Mặt khác: \(3\left(x-2\right)^2+6\left(y-1\right)^2+22=0\)

Suy ra: Không có giá trị nào của x; y thoả mãn yêu cầu đề bài.

#Ayumu

a) 5x² + 10y² - 6xy - 4x - 2y + 3 

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

= ( x - 3y )² + ( 2x - 1 )² + ( y - 1 )² + 1 ≥ 1 > 0 ∀ x, y, z

=> đpcm 

b) x² + 4y² + z² - 2x - 6z + 8y + 15 

= ( x² - 2x + 1 ) + ( 4y² + 8y + 4 ) + ( z² - 6z + 9 ) + 1

= ( x - 1 )² + ( 2y + 2 )² + ( z - 3 )² + 1 ≥ 1 > 0 ∀ x, y, z

=> đpcm

\(4x^2+3y^2-4x+30y+78=0\)

=>\(\left(4x^2-4x+1\right)+3\left(y^2+10y+25\right)+2=0\)

=>\(\left(2x-1\right)^2+3\left(y+5\right)^2+2=0\)(vô lý)

=>\(\left(x,y\right)\in\varnothing\)

Chứng minh nó luôn > 0 là xong 

\(3x^2+y^2+10x-2xy+26=0\)

\(\left(x-y\right)^2+2x^2+10x+26=0\)

\(\left(x-y\right)^2+\left(2x^2+10x+\frac{5\sqrt{2}}{2}^2\right)+\frac{27}{2}=0\)

\(\left(x-y\right)^2+\left(\sqrt{2}x+\frac{5\sqrt{2}}{2}\right)^2+\frac{27}{2}\ge\frac{27}{2}>0\)

vậy ko có giá trị xy thỏa mã đt