\(x^4+6x^3+7x^2-6x+1=x^4+3x^3-x^2+3x^3+9x^2-3x-x^2-3x+1\)

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

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

Trả lời:

\(C=x^2+10y^2+6xy-y+1\)

\(C=x^2+9y^2+y^2+6xy-y+\frac{1}{4}+\frac{3}{4}\)

\(C=\left(x^2+6xy+9y^2\right)+\left(y^2-y+\frac{1}{4}\right)+\frac{3}{4}\)

\(C=\left(x+3y\right)^2+\left(y-\frac{1}{2}\right)^2+\frac{3}{4}\ge\frac{3}{4}\forall x;y\)

Dấu "=" xảy ra khi \(\hept{\begin{cases}x+3y=0\\y-\frac{1}{2}=0\end{cases}\Leftrightarrow\hept{\begin{cases}x=-\frac{3}{2}\\y=\frac{1}{2}\end{cases}}}\)

Vậy GTNN của C = 3/4 khi \(x=-\frac{3}{2};y=\frac{1}{2}\)

Trả lời:

a, ( x + y )² + ( x - y )² - 2x² = x² + 2xy + y² + x² - 2xy + y² - 2x² = 2y² 

b, 2( x - y )( x + y ) + ( x + y )² + ( x - y )² 

= 2( x² - y² ) + x² + 2xy + y² + x² - 2xy + y² 

= 2x² - 2y² + x² + 2xy + y² + x² - 2xy + y² 

= 4x² 

c, ( x - 3 )( x + 3 ) - ( x - 5 )

= x² - 9 - x + 5

= x² - x - 4

d, ( 2x + 1 )² + 2( 2x + 1 )( 3x - 1 ) + ( 3x - 1 )²

= 4x² + 4x + 1 + ( 4x + 2 )( 3x - 1 ) + 9x² - 6x + 1

= 4x² + 4x + 1 + 12x² - 4x + 6x - 2 + 9x² - 6x + 1 

= 25x² 

e, ( 3x + 5 )² - 2( 3x + 5 )( 2x + 5 ) + ( 2x + 5 )²

= 9x² + 30x + 25 + ( - 6x - 10 )( 2x + 5 ) + 4x² + 20x + 25

= 9x² + 30x + 25 - 12x² - 30x - 20x - 50 + 4x² + 20x + 25

= x² 

\(a+b+c+d=0\Leftrightarrow b+c=-\left(a+d\right)\Leftrightarrow\left(b+c\right)^3=-\left(a+d\right)^3\)

\(a^3+b^3+c^3+d^3=\left(b+c\right)^3-3bc\left(b+c\right)+\left(a+d\right)^3-3ad\left(a+d\right)\)

\(=-3bc\left(b+c\right)+3ad\left(b+c\right)\)

\(=3\left(b+c\right)\left(ad-bc\right)\)

C . x² - y²

Trả lời:

Ta có: ( x - y )( x + y ) = x² - y²

=> Chọn C