x² + x + 1/4 = ( x + 1/2 )²

Trả lời:

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

...

1) =\(x^7-x+x^2+x\)+1

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

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

=x(x^3+1)(x-1)(x^2+x+1)+(x^2+x+1)

=[(x^4+x)(x-1)+1](x^2+x+1)

=(x^5-x^4+x^2-x)(x^2+x+1)

Trả lời:

1, x⁷ + x² + 1 

= x⁷ + x² + 1 + x⁶ - x⁶ + x⁵ - x⁵ + x⁴ - x⁴ + x³ - x³ + x² - x² + x - x 

= ( x⁷ + x⁶ + x⁵ ) - ( x⁶ + x⁵ + x⁴ ) + ( x⁴ + x³ + x² ) - ( x³ + x² + x ) + ( x² + x + 1 )

= x⁵ ( x² + x + 1 ) - x⁴ ( x² + x + 1 ) + x² ( x² + x + 1 ) - x ( x² + x + 1 ) + ( x² + x + 1 )

= ( x² + x + 1 )( x⁵ - x⁴ + x² - x + 1 )

b, x⁸ + x⁷ + 1 

= x⁸ + x⁷ + 1 + x⁶ - x⁶ + x⁵ - x⁵ + x⁴ - x⁴ + x³ - x³ + x² - x² + x - x 

= ( x⁸ + x⁷ + x⁶ ) - ( x⁶ + x⁵ + x⁴ ) + ( x⁵ + x⁴ + x³ ) - ( x³ + x² + x ) + ( x² + x + 1 ) 

= x⁶ ( x² + x + 1 ) - x⁴ ( x² + x + 1 ) + x³ ( x² + x + 1 ) - x ( x² + x + 1 ) + ( x² + x + 1 )

= ( x² + x + 1 )( x⁶ - x⁴ + x³ - x + 1 )

Trả lời:

a, 3x²y - 6xy = 3xy ( x - 2 )

b, x² - y² - 9x + 9y 

= ( x² - y² ) - ( 9x - 9y )

= ( x - y )( x + y ) - 9 ( x - y )

= ( x - y )( x + y - 9 )

c, x³ - 6x² - y²x + 9x 

= x ( x² - 6x - y² + 9 )

= x [ ( x² - 6x + 9 ) - y² ]

= x [ ( x - 3 )² - y² ]

= x ( x - 3 - y )( x - 3 + y )

3x²y - 6xy = 3xy( x - 2 )

x² - y² - 9x + 9y = ( x - y )( x + y ) - 9( x - y ) = ( x - y )( x + y - 9 )

x³ - 6x² - y²x + 9x = x( x² - 6x - y² + 9 ) = x[ ( x - 3 )² - y² ] = x( x - y - 3 )( x + y - 3 )

Trả lời:

a, \(-xy.\left(x^2+2xy-3\right)=-x^3y-2x^2y^2+3xy\)

b, \(\left(12x^6y^5-3x^3y^4+4x^2y\right):6x^2y\)

\(=12x^6y^5:6x^2y^2-3x^3y^4:6x^2y+4x^2y+6x^2y\)

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

a.\(\left(-xy\right)\left(x^2+2xy-3\right)=-x^3y-2x^2y^2+6xy\)

b.\(\left(12x^6y^5-3x^3y^4+4x^2y\right):6x^2y=2x^4y^4-\frac{1}{2}xy^3+\frac{2}{3}\)

Mn giúp mk vs, mk cần gấp á

Thanks!

Giải phương trình: x^4-5x^3+6x^2-5x+1=0

x=2-căn bậc hai(3),

x=căn bậc hai(3)+2;

x = -(căn bậc hai(3)*i-1)/2;

x = (căn bậc hai(3)*i+1)/2;

Trả lời:

2ab⁴ - 8ab³ + 8ab² 

= 2ab² ( b² - 4b + 4 )

= 2ab² ( b - 2 )² 

\(\frac{1}{a+2b+3c}+\frac{1}{2a+3b+c}+\frac{1}{3a+b+2c}\)

\(\le\frac{1}{9}\left(\frac{1}{a+c}+\frac{1}{b+c}+\frac{1}{b+c}+\frac{1}{a+b}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+b}+\frac{1}{a+c}+\frac{1}{a+c}\right)\)

\(\le\frac{1}{36}\left(\frac{1}{a}+\frac{1}{c}+\frac{1}{b}+\frac{1}{c}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a}+\frac{1}{b}+\frac{1}{a}+\frac{1}{b}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a}+\frac{1}{b}+\frac{1}{a}+\frac{1}{c}+\frac{1}{a}+\frac{1}{c}\right)\)

\(=\frac{1}{6}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\)

\(=\frac{1}{6}\left(\frac{a+b+c}{abc}\right)=\frac{1}{6}\)

Dấu \(=\)khi \(a=b=c=3\).

bài này áp dụng bđt phụ \(\frac{1}{x_1+x_2+x_3+x_4+x_5+x_6}\le\frac{1}{36}\left(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\frac{1}{x_4}+\frac{1}{x_5}+\frac{1}{x_6}\right)\)

là oke nhé =)) 

về cách chứng minh thì bạn dùng svacxo cho vp là xong 

A = ( 4x + 1 )² + 2 ≥ 2 > 0 ∀ x ( đpcm )

B = ( y - 5/2 )² + 7/4 ≥ 7/4 > 0 ∀ x ( đpcm )

C = 2( x - 1/2 )² + 3/2 ≥ 3/2 > 0 ∀ x ( đpcm )

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

Trả lời:

a, \(A=16x^2+8x+3=\left(16x^2+8x+1\right)+2=\left(4x+1\right)^2+2\ge2>0\forall x\) 

Dấu "=" xảy ra khi x = - 1/4

Vậy bt A luôn dương với mọi x.

b, \(B=y^2-5y+8=x^2-2.y.\frac{5}{2}+\frac{25}{4}+\frac{7}{4}=\left(x-\frac{5}{2}\right)^2+\frac{7}{4}\ge\frac{7}{4}>0\forall y\)

Dấu "=" xảy ra khi x = 5/2

Vậy bt B luôn dương với mọi y.

c, 

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

\(=2\left[\left(x-\frac{1}{2}\right)^2+\frac{3}{4}\right]=2\left(x-\frac{1}{2}\right)^2+\frac{3}{2}\ge\frac{3}{2}>0\forall x\)

Dấu "=" xảy ra khi x = 1/2

Vậy bt C luôn dương với mọi x.

d, \(D=9x^2-6x+25y^2+10y+4\)

\(=9x^2-6x+25y^2+10y+1+1+2\)

\(=\left(9x^2-6x+1\right)+\left(25y^2+10y+1\right)+2\)

\(=\left(3x-1\right)^2+\left(5y+1\right)^2+2\ge2>0\forall x;y\)

Dấu "=" xảy ra khi x = 1/3; y = - 1/5

Vậy bt D luôn dương với mọi x;y

Trả lời:

\(A=\left(1-\frac{x^2-x}{x-1}\right)\left(1+\frac{x^2+x}{x+1}\right)+x^2\) \(\left(ĐKXĐ:x\ne\pm1\right)\)

\(=\frac{x-1-x^2+x}{x-1}.\frac{x+x+x^2+x}{x+1}+x^2\)

\(=\frac{-x^2+2x-1}{x-1}.\frac{x^2+2x+1}{x+1}\)\(+x^2\)

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

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

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

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

\(B=\left(2-\frac{x^2-x}{x-1}\right)\left(2+\frac{x^2+x}{x+1}\right)\) \(\left(ĐKXĐ:x\ne\pm1\right)\)

\(=\frac{2x-2-x^2+x}{x-1}.\frac{2x+2+x^2+x}{x+1}\)

\(=\frac{-x^2+3x-2}{x-1}.\frac{x^2+3x+2}{x+1}\)

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

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

\(=\frac{-\left[x\left(x-1\right)-2\left(x-1\right)\right]}{x-1}.\frac{x\left(x+1\right)+2\left(x+1\right)}{x+1}\)

\(=\frac{-\left(x-1\right)\left(x-2\right)}{x-1}.\frac{\left(x+1\right)\left(x+2\right)}{x+1}\)

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

\(a.\left(x^2+2x+x+2\right)\left(x^2+5x+6x+30\right)-5\)

\(=\left(x+1\right)\left(x+2\right)\left(x+5\right)\left(x+6\right)-5=\left(x^2+7x+6\right)\left(x^2+7x+10\right)\)

Đặt \(x^2+7x+8=a\Rightarrow\text{Biểu thức }=\left(a-2\right)\left(a+2\right)-5=a^2-9=\left(a-3\right)\left(a+3\right)\)

nên : \(BT=\left(x^2+7x+5\right)\left(x^2+7x+11\right)\)

b.\(BT=\left(x^2+5ax+4a^2\right)\left(x^2+5ax+6a^2\right)+a^4\)

Đặt \(x^2+5ax+5a^2=y\Rightarrow BT=\left(y-a^2\right)\left(y+a^2\right)+a^4=y^2=\left(x^2+5ax+5a^2\right)^2\)