\(A=-\left(x^2-6x+9\right)=-\left(x-3\right)^2\)

Do \(\left(x-3\right)^2\ge0;\forall x\Rightarrow-\left(x-3\right)^2\le0;\forall x\)

\(\Rightarrow A\le0\Rightarrow A_{max}=0\) khi \(x=3\)

\(B=4x^2-4x+1+14=\left(2x-1\right)^2+14\)

Do \(\left(2x-1\right)^2\ge0;\forall x\Rightarrow\left(2x-1\right)^2+14\ge14;\forall x\)

\(\Rightarrow B_{min}=14\) khi \(2x-1=0\Rightarrow x=\dfrac{1}{2}\)

a.

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

b.

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

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

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

\(=\left(4x\right)^2=16x^2\)

mình cần gấp áa

a: X,Y trái dấu

=>XY<0

=>\(-2abc^3\cdot3a^2b^3c^5< 0\)

=>\(-6a^3b^4c^8< 0\)

=>\(a^3>0\)

=>a>0

b: X,Y cùng dấu

=>X*Y>0

=>\(-2abc^3\cdot3a^2b^3c^5>0\)

=>\(-6a^3b^4c^8>0\)

=>\(a^3< 0\)

=>a<0

c: \(X\cdot Y=-5a^2n\cdot b\cdot3a^4n\cdot b^5=-15a^6n^2b^6< =0\forall a,b,n\)

=>X và Y không thể cùng có giá trị âm

Đề là \(\left(1+\dfrac{1}{a}\right)\left(1+\dfrac{1}{b}\right)\ge9\) với đúng chứ em?

thầy trả lời giúp em với

\(\Leftrightarrow\left(6x^2+2xy-8x\right)+\left(3xy+y^2-4y\right)+\left(3x+y-4\right)=1\)

\(\Leftrightarrow2x\left(3x+y-4\right)+y\left(3x+y-4\right)+\left(3x+y-4\right)=1\)

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

Ta có bảng sau:

Vậy \(\left(x;y\right)=\left(5;-12\right);\left(5;-10\right)\)

 A = (14⁸)²⁰²⁰ + 10 

A =  (14⁸)^5.404 + 10

A = (14⁵)^8.404 + 10

A = 537824³²³² + 10

537824 \(\equiv\) 1 (mod 11)

537824³²³² \(\equiv\) 1³²³² (mod 11) \(\equiv\) 1 (mod 11)

10 \(\equiv\) 10 (mod 11)

⇒ 537824³²³² + 10  \(\equiv\) 1 + 10 (mod 11)

⇒537824³²³² + 10 \(\equiv\) 11 (mod 11) \(\equiv\) 0 (mod 11)

⇒ A = (14⁸)²⁰²⁰ + 10 \(⋮\) 11 (đpcm)

\(14\equiv3\left(mod11\right)\Rightarrow\left(14^8\right)^{2020}\equiv\left(3^8\right)^{2020}\left(mod11\right)\)

\(\left(3^8\right)^{2020}=3^{8.404.5}=\left(3^5\right)^{3232}=\left(243\right)^{3232}\)

\(243\equiv1\left(mod11\right)\Rightarrow243^{3232}\equiv1\left(mod11\right)\)

\(\Rightarrow\left(14^8\right)^{2020}\equiv1\left(mod11\right)\)

\(\Rightarrow\left(14^8\right)^{2020}+10⋮11\)

Đặt \(P=-x^2+4xy-5y^2-2x+4y-5\)

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

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

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

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

Do \(\left\{{}\begin{matrix}-\left(x-2y+1\right)^2\le0\\-y^2\le0\\-4< 0\end{matrix}\right.\) ; \(\forall x;y\)

\(\Rightarrow-\left(x-2y+1\right)^2-y^2-4< 0;\forall x;y\)

Vậy P luôn âm

`a^3 + b^3 + c^3 = 3abc`

`=> a^3 + b^3 + c^3 - 3abc = 0`

`=> (a+b)^3 - 3ab(a+b) + c^3 - 3abc = 0`

`=> ((a+b)^3  + c^3) - (3ab(a+b) + 3abc) = 0`

`=> (a+b+c) ((a+b)^2 - (a+b)c + c^2) - 3ab(a+b+c) = 0`

`=> (a+b+c)(a^2 + 2ab + b^2 - ac - bc + c^2) - 3ab(a+b+c) = 0`

`=> (a+b+c)(a^2 + 2ab + b^2 - ac - bc + c^2 - 3ab) = 0`

`=> (a+b+c)(a^2 - ab + b^2 - ac - bc + c^2) = 0`

Trường hợp 1: 

`a+b+c = 0 (đpcm)`

Trường hợp 2: 

`a^2 - ab + b^2 + ac + bc + c^2 = 0`

`<=> 2a^2 - 2ab + 2b^2 - 2bc +2c^2 - 2ca = 0`

`<=> a^2 - 2ab + b^2 + b^2 - 2bc +c^2 + c^2 - 2ac + a^2 = 0`

`<=> (a-b)^2 + (b-c)^2 + (c-a)^2 = 0`

Do `{((a-b)^2 >=0),((b-c)^2 >=0),((c-a)^2 >=0):}`

`=> (a-b)^2 + (b-c)^2 + (c-a)^2 >= 0`

Dấu = có khi: 

`{(a=b),(b=c),(c=a):}`

Hay `a=b=c  (đpcm)`

Ta có :a^3+b^3+c^3=3abc⇒a^3+b^3+c^3-3abc=0

⇒(a+b+c)(a^2+b^2+c^2-ab-bc-ac)=0

TH1: a+b+c=0

TH2:a^2+b^2+c^2-ab-ac-bc=0

⇒2a^2+2b^2+2c^2-2ab-2bc-2ac=0

(a-b)^2+(b-c)^2+(c-a)^2=0

⇒a=b=c