\(y\in\left(-\infty;\infty\right)\)

\(-2y^2-3xy-2y+2x^2+6x=1\)

\(-2y^2-3xy-2y-2x^2+6x-1=0\)

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

\(y=\frac{\sqrt{25x^2+60x-4-3x-2}}{4}\)

\(y=-\frac{\sqrt{25x^2+60x-4+3x+2}}{4}\)

#Ứng Lân

Áp dụng Bunyakovsky, ta có :

\(\left(1+1\right)\left(x^2+y^2\right)\ge\left(x.1+y.1\right)^2=1\)

=> \(\left(x^2+y^2\right)\ge\frac{1}{2}\)

=> \(Min_C=\frac{1}{2}\Leftrightarrow x=y=\frac{1}{2}\)

Mấy cái kia tương tự 

a) cho A(x) = 0

\(=>2x^2-4x=0\)

\(x\left(2-4x\right)=0\Rightarrow\left[{}\begin{matrix}x=0\\4x=2\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=0\\x=\dfrac{1}{2}\end{matrix}\right.\)

b)\(B\left(y\right)=4y-8\)

cho B(y) = 0

\(4y-8=0\Rightarrow4y=8\Rightarrow y=2\)

c)\(C\left(t\right)=3t^2-6\)

cho C(t) = 0

\(=>3t^2-6=0=>3t^2=6=>t^2=2\left[{}\begin{matrix}t=\sqrt{2}\\t=-\sqrt{2}\end{matrix}\right.\)

d)\(M\left(x\right)=2x^2+1\)

cho M(x) = 0

\(2x^2+1=0\Rightarrow2x^2=-1\Rightarrow x^2=-\dfrac{1}{2}\left(vl\right)\)

vậy M(x) vô nghiệm

e) cho N(x) = 0

\(2x^2-8=0\)

\(2\left(x^2-4\right)=0\)

\(2\left(x^2+2x-2x-4\right)=0\)

\(2\left(x-2\right)\left(x+2\right)=0\)

\(\Rightarrow\left[{}\begin{matrix}x-2=0\\x+2=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=2\\x=-2\end{matrix}\right.\)

`e, N(x) = 2x^2 - 8 = 2( x^2 - 4 ) = 2( x-2 )( x + 2 )`

Xét `N(x)=0`

`=> 2(x-2)(x+2)=0`

`=>(x-2)(x+2)=0`

`=>x-2=0` hoặc `x+2=0`

`=>x=2` hoặc `x=-2`

Vậy `x in { +-2 }` là nghiệm của `N(x)` 

Ta có \(2y^2⋮2\Rightarrow x^2\equiv1\left(mod2\right)\Rightarrow x^2\equiv1\left(mod4\right)\Rightarrow2y^2⋮4\Rightarrow y⋮2\Rightarrow x^2\equiv5\left(mod8\right)\) (vô lí).

Vậy pt vô nghiệm nguyên.

2: \(PT\Leftrightarrow3x^3+6x^2-12x+8=0\Leftrightarrow4x^3=\left(x-2\right)^3\Leftrightarrow\sqrt[3]{4}x=x-2\Leftrightarrow x=\dfrac{-2}{\sqrt[3]{4}-1}\).

\(\Leftrightarrow\left(x^2-2xy+y^2\right)+\left(4x^2-4x+1\right)+\left(y^2-2y+1\right)< 3\)

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

\(\Rightarrow\left(2x-1\right)^2< 3\) (1)

\(\Rightarrow\left(2x-1\right)^2=\left\{0;1\right\}\)

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

- Với \(x=0\Rightarrow2y^2-2y< 1\Rightarrow\left(2y-1\right)^2< 3\Rightarrow\left[{}\begin{matrix}y=0\\y=1\end{matrix}\right.\) (giải như (1))

- Với \(x=1\Rightarrow2y^2+5< 4y+5\Rightarrow y^2-2y< 0\)

\(\Rightarrow y\left(y-2\right)< 0\Rightarrow0< y< 2\Rightarrow y=1\)

Vậy \(\left(x;y\right)=\left(0;0\right);\left(0;1\right);\left(1;1\right)\)

Để hệ phương trình có nghiệm duy nhất thì \(\dfrac{m+1}{m^2}\ne\dfrac{-2}{-1}=2\)

=>\(2m^2\ne m+1\)

=>\(2m^2-m-1\ne0\)

=>\(\left(m-1\right)\left(2m+1\right)\ne0\)

=>\(m\notin\left\{1;-\dfrac{1}{2}\right\}\)

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

=>\(\left\{{}\begin{matrix}\left(m+1\right)x-2y=m-1\\2m^2\cdot x-2y=2m^2+4m\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x\left(2m^2-m-1\right)=2m^2+4m-m+1\\\left(m+1\right)x-2y=m-1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x\cdot\left(m-1\right)\left(2m+1\right)=2m^2+3m+1=\left(m+1\right)\left(2m+1\right)\\\left(m+1\right)x-2y=m-1\end{matrix}\right.\)

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

=>\(\left\{{}\begin{matrix}x=\dfrac{m+1}{m-1}\\2y=\dfrac{m^2+2m+1-\left(m-1\right)^2}{m-1}\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}x=\dfrac{m+1}{m-1}\\y=\dfrac{m^2+2m+1-m^2+2m-1}{2m-2}=\dfrac{4m}{2m-2}=\dfrac{2m}{m-1}\end{matrix}\right.\)

Để x,y đều nguyên thì \(\left\{{}\begin{matrix}m+1⋮m-1\\2m⋮m-1\end{matrix}\right.\)

=>\(\left\{{}\begin{matrix}m-1+2⋮m-1\\2m-2+2⋮m-1\end{matrix}\right.\)

=>\(2⋮m-1\)

=>\(m-1\in\left\{1;-1;2;-2\right\}\)

=>\(m\in\left\{2;0;3;-1\right\}\)

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

\(\Leftrightarrow\left\{{}\begin{matrix}\left(m+1\right)x-2y=m-1\\2m^2x-2y=2m^2+4m\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(2m^2-m-1\right)x=2m^2+3m+1\\y=m^2x-m^2-2m\end{matrix}\right.\)

Pt có nghiệm duy nhất khi \(2m^2-m-1\ne0\Rightarrow m\ne\left\{1;-\dfrac{1}{2}\right\}\)

Khi đó: \(\left\{{}\begin{matrix}x=\dfrac{2m^2-2m-1}{2m^2+3m+1}=\dfrac{\left(m-1\right)\left(2m+1\right)}{\left(m+1\right)\left(2m+1\right)}=\dfrac{m-1}{m+1}\\y=m^2x-m^2-2m=\dfrac{-4m^2-2m}{m+1}\end{matrix}\right.\)

Để x nguyên \(\Rightarrow\dfrac{m-1}{m+1}\in Z\Rightarrow1-\dfrac{2}{m+1}\in Z\)

\(\Rightarrow\dfrac{2}{m+1}\in Z\)

\(\Rightarrow m+1=Ư\left(2\right)=\left\{-2;-1;1;2\right\}\)

\(\Rightarrow m=\left\{-3;-2;0;1\right\}\)

Thay vào y thấy đều thỏa mãn y nguyên.

Vậy ...

1) \(\left(x+1\right)^2=x^2+2x+1\)

2) \(\left(2x+1\right)^2=4x^2+4x+1\)

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

4) \(\left(2x+3\right)^2=4x^2+12x+9\)

5) \(\left(3x+2y\right)^2=9x^2+12xy+4y^2\)

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

7) \(\left(x^3+1\right)^2=x^6+2x^3+1\)

8) \(\left(x^2+y^3\right)^2=x^4+2x^2y^3+y^6\)

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

10) \(\left(\dfrac{1}{2}x+\dfrac{1}{3}y\right)^2=\dfrac{1}{4}x^2+\dfrac{1}{3}xy+\dfrac{1}{9}y^2\)

pt : x^3.y^3+x^3 = 91

<=> x^3.(y^3-1) = 91

Đến đó bạn dùng ước bội mà giải nha

Tk mk