(u-1)^2 + (v+3/2)^2 + 11,75 \(\ge\)11,75

''='' <=> u = 1, v = -3/2

=> Min = 11,75 <=> u = 1, v = -3/2

Đặt \(A=u^2+v^2-2u+3v+15\)

\(=\left(u^2-2u+1\right)+\left(v^2+3v+\frac{9}{16}\right)+\frac{215}{16}\)

\(=\left(u-1\right)^2+\left(v+\frac{3}{4}\right)^2+\frac{215}{16}\ge\frac{215}{16}\)

Dấu "=" xảy ra khi u = 1, v = -3/4

Vậy Amin = 215/16 khi u = 1, v = -3/4

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

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

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

= (a-b)(a+b)(c-a) + (c-a)(c+a)(b-a)

= (a-b)(c-a)(a+b-c-a) = (a-b)(b-c)(c-a)

A = 8x^3 - 12x^2 + 18x - 12x^2 + 18x - 27 + 24x^2 - 16x

= 8x^3 + 20x - 27

Đặt \(x^2+y^2=a;y^2+z^2=b\)

\(\Rightarrow z^2-x^2=\left(y^2+z^2\right)-\left(x^2+y^2\right)=b-a\)

\(\Rightarrow A=a^3+\left(b-a\right)^3-b^3\)

\(=\left(a-b\right)\left(a^2+ab+b^2\right)-\left(a-b\right)^3\)

\(=\left(a-b\right)\left[a^2+ab+b^2-a^2+2ab-b^2\right]\)

\(=3ab\left(a-b\right)=3\left(x^2+y^2\right)\left(y^2+z^2\right)\left(x^2-z^2\right)\)

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

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

\(B=x^3+y^3+z^3+3.\left(x+y\right)\left(y+z\right)\left(z+x\right)-x^3-y^3-z^3\)

\(B=3.\left(x+y\right)\left(y+z\right)\left(z+x\right)\)

Đây là hằng đẳng thức:

\(\left(a+b+c\right)^3=a^3+b^3+c^3+3\left(a+b\right)\left(b+c\right)\left(c+a\right)\)

a) 5xy ( x - y ) - 2x + 2y

= 5xy ( x - y ) - 2 ( x - y )

= ( x - y ) ( 5xy - 2 )

b) 6x-2y-x(y-3x)

= 2 ( y - 3x ) - x ( y - 3x )

= ( y - 3x ( ( 2 - x )

c)  x² + 4x - xy-4y

= x ( x + 4 ) - y ( x + 4 )

( x + 4 ) ( x - y )

d) 3xy + 2z - 6y - xz 

= ( 3xy - 6y ) + ( 2z - xz )

= 3y ( x - 2 ) + z ( x - 2 )

= ( x - 2 ) ( 3y + z )

a,5xy(x-y)-2x+2y=5xy(x-y)-2(x-y)=(x-y)(5xy-2)

b,6x-2y-x(y-3x)=-2(y-3x)-x(y-3x)=(y-3x)(-2-x)

c,x^2+4x-xy-4y=x(x+4)-y(x+4)=(x+4)(x-y)

d,3xy+2z-6y-xz=(3xy-6y)+(2z-xz)=3y(x-2)+z(2-x)=3y(x-2)-z(x-2)=(x-2)(3y-z)

11)

a,4-9x^2=0

(2-3x)(2+3x)=0

2-3x=0=>x=2/3 hoặc 2+3x=0=>x=-2/3

b,x^2 +x+1/4=0

(x+1/2)^2 =0

x+1/2=0

x=-1/2

c,2x(x-3)+(x-3)=0

(x-3)(2x+1)=0

x-3=0=>x=3 hoặc 2x+1=0=>x=-1/2

d,3x(x-4)-x+4=0

3x(x-4)-(x-4)=0

(x-4)(3x-1)=0

x-4=0=>x=4 hoặc 3x-1=0=>x=1/3

e,x^3-1/9x=0

x(x^2-1/9)=0

x(x+1/3)(x-1/3)=0

x=0 hoặc x+1/3=0=>x=-1/3 hoặc x-1/3=0=>x=1/3

f,(3x-y)^2-(x-y)^2 =0

(3x-y-x+y)(3x-y+x-y)=0

2x(4x-2y)=0

4x(2x-y)=0

x=0hoặc 2x-y=0=>x=y/2

\(a,\left(a+b\right)^3-\left(a-b\right)^3-6a^2b\)

\(=a^3+3a^2b+3ab^2+b^3-a^3+3a^2b-3ab^2+b^3-6a^2b\)

\(=2b^3\)

\(b,\left(a+b\right)^3+\left(a-b\right)^3-6ab^2\)

\(=a^3+3a^2b+3ab^2+b^3+a^3-3a^2b+3ab^2-b^3-6ab^2\)

\(=2a^3\)

ai lam nhanh nhat thi minh k cho