\(\)\(x^2+5y^2+2x+6y+34\)
\(=\left(x^2+2x.1+1\right)+\left[\left(\sqrt{5}y\right)^2+2.\sqrt{5}x.\dfrac{3}{\sqrt{5}}+\left(\dfrac{3}{\sqrt{5}}\right)^2\right]+\dfrac{156}{5}\)
\(=\left(x+1\right)^2+\left(\sqrt{5}y+\dfrac{3}{\sqrt{5}}\right)^2+\dfrac{156}{5}\)
Vì \(\left\{{}\begin{matrix}\left(x+1\right)^2\text{≥ 0}\\\left(\sqrt{5}y+\dfrac{3}{\sqrt{5}}\right)^2\text{≥ 0}\end{matrix}\right.\)
⇒  \(\left(x+1\right)^2+\left(\sqrt{5}y+\dfrac{3}{\sqrt{5}}\right)^2+\dfrac{156}{5}\text{ }\text{≥}\dfrac{156}{5}\)
⇒ đa thức vô nghiệm

a/

\(\Leftrightarrow\left(x^2+4y^2+1-4xy+2x-4y\right)+\left(y^2-6y+9\right)-19=0\)

\(\Leftrightarrow\left(x-2y+1\right)^2+\left(y-3\right)^2=19\)

Do 19 không thể phân tích thành tổng của 2 số chính phương nên pt vô nghiệm

b/

\(\left(4x^2+4y^2+8xy\right)+\left(x^2-2x+1\right)+\left(y^2+2y+1\right)=0\)

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

Do x; y nguyên dương nên \(\left(2x+2y\right)^2>0\Rightarrow VT>0\)

Pt vô nghiệm

c/

\(\Leftrightarrow\left(x^2+4y^2+25-4xy+10x-20y+25\right)+\left(y^2-2y+1\right)+\left|x+y+z\right|=0\)

\(\Leftrightarrow\left(x-2y+5\right)^2+\left(y-1\right)^2+\left|x+y+z\right|=0\)

Do x;y;z nguyên dương nên \(\left|x+y+z\right|>0\Rightarrow VT>0\)

Vậy pt vô nghiệm

d/

\(\Leftrightarrow\left(x^2+y^2+z^2+2xy+2yz+2zx\right)+\left(x^2+10x+25\right)+\left(y^2+6y+9\right)=0\)

\(\Leftrightarrow\left(x+y+z\right)^2+\left(x+5\right)^2+\left(y+3\right)^2=0\)

Do x;y;z nguyên dương nên vế phái luôn dương

Pt vô nghiệm

x² - 8x + 20

= x² - 8x + 20

= ( x² - 8x + 16 ) + 4

= ( x - 4 )² + 4 ≥ 4 > 0 ∀ x ( đpcm )

x² + 5y² + 2x + 6y + 34

x² + 5y² + 2x + 6y + 34

= ( x² + 2x + 1 ) + ( 5y² + 6y + 9/5 ) + 156/5

= ( x + 1 )² + 5( y² + 6/5y + 9/25 ) + 156/5

= ( x + 1 )² + 5( y + 3/5 )² + 156/5 ≥ 156/5 > 0 ∀ x, y ( đpcm )

a) \(x^2-8x+20\)

\(=x^2-2.x.4+16+4\)

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

Có: \(\left(x-4\right)^2\ge0\Rightarrow\left(x-4\right)^2+4>0\)

Hay:.............

b) \(x^2+11\)

Có: \(x^2\ge0\Rightarrow x^2+11>0\)

Hay:.............

c) \(4x^2-12x+11\)

\(=4\left(x^2-3x+\frac{11}{4}\right)\)

\(=4\left(x^2-2.x.\frac{3}{2}+\frac{9}{4}+\frac{1}{2}\right)\)

\(=4\left(x-\frac{3}{2}\right)^2+2>0\)

d) \(x^2+5y^2+2x+6y+34\)

\(=x^2+2.x.1+1+y^2+4y^2+2.y.3+9+24\)

\(=\left(x^2+2.x.1+1\right)+\left(y^2+2.y.3+9\right)+4y^2+24\)

\(=\left(x+1\right)^2+\left(y+3\right)^2+\left(2y\right)^2+24\)

Ta có: \(\left\{{}\begin{matrix}\left(x+1\right)^2\ge0\\\left(y+3\right)^2\ge0\\\left(2y\right)^2\ge0\end{matrix}\right.\)

\(\Rightarrow\left(x+1\right)^2+\left(y+3\right)^2+\left(2y\right)^2+24>0\)

f) \(x^2-2x+y^2+4y+6\)

\(=x^2-2.x.1+1+y^2+2.y.2+4+1\)

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

Lời giải:
a. Thay $y=x+1$ vào điều kiện ban đầu có:

$3x+5(x+1)=13$
$8x+5=13$

$8x=8$

$x=1$

$y=x+1=2$
b. Thay $x=y+5$ vô điều kiện đầu thì:

$2(y+5)-3y=4$

$-y+10=4$

$-y=-6$

$y=6$

$x=6+5=11$

c. Thay $y=x-2$ vô điều kiện đầu thì:

$-x+5(x-2)=-6$

$4x-10=-6$

$4x=10+(-6)=4$

$x=1$

$y=x-2=1-2=-1$

a) Ta có: \(\left\{{}\begin{matrix}3x+5y=13\\x+1=y\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}3x+5y=13\\x-y=-1\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}3x+5y=13\\3x-3y=-3\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}8y=16\\x+1=y\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=2\\x=y-1=2-1=1\end{matrix}\right.\)

b) Ta có: \(\left\{{}\begin{matrix}2x-3y=4\\x=y+5\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}2x-3y=4\\x-y=5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}2x-3y=4\\2x-2y=10\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}-y=-6\\x=y+5\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}y=6\\x=11\end{matrix}\right.\)

c) Ta có: \(\left\{{}\begin{matrix}-x+5y=-6\\y=x-2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}-x+5y=-6\\x-y=2\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}4y=-4\\y=x-2\end{matrix}\right.\)

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

\(a,=2x^2-7xy-30y^2+30x^2+2xy=32x^2-5xy-30y^2\\ b,=x^2-10x+25+2x^2-8=3x^2-10x+17\\ c,=x^3+8-x^3+5=13\\ d,=x^3-x^2+x-x^2+x-1+x^2-1=x^3-x^2+2x-2\)

a)

\(\dfrac{x}{5}=\dfrac{y}{2}=\dfrac{3x-2y}{3.5-2.2}=\dfrac{-55}{11}=-5\)

=> \(\left\{{}\begin{matrix}x=-5.5=-25\\y=-5.2=-10\end{matrix}\right.\)

b)

\(\dfrac{x}{3}=\dfrac{y}{2}=\dfrac{2x+5y}{2.3+5.2}=\dfrac{48}{16}=3\)

=> \(\left\{{}\begin{matrix}x=3.3=9\\y=3.2=6\end{matrix}\right.\)

c)

Có: \(\dfrac{x}{y}=-\dfrac{5}{2}\Leftrightarrow-\dfrac{x}{5}=\dfrac{y}{2}=\dfrac{x+y}{-5+2}=\dfrac{30}{-3}=-10\)

=> \(\left\{{}\begin{matrix}x=-10.-5=50\\y=-10.2=-20\end{matrix}\right.\)

d)

Có: \(\dfrac{x}{y}=\dfrac{4}{3}\Leftrightarrow\dfrac{x}{4}=\dfrac{y}{3}=\dfrac{2x+3y}{2.4+3.3}=\dfrac{34}{17}=2\)

=> \(\left\{{}\begin{matrix}x=2.4=8\\y=2.3=6\end{matrix}\right.\)

\(A=5x^2-3x-x^3+x^2+x^3-62x-10+3x\\ A=6x^2-62x-10\\ B=x^3+x^2+x-x^3-x^2-x+5=5\\ C=3x^2y-15xy^2+15xy^2-10y^3+10y^2-3x^2y-4=-4\)

b: Ta có: \(B=x\left(x^2+x+1\right)-x^2\left(x+1\right)-x+5\)

\(=x^3+x^2+x-x^3-x^2-x+5\)

=5