\(4x^2y-3xy^2+7xy-4x^2y-6xy+x-y\)

\(=4x^2y-4x^2y-3xy^2+7xy-6xy+x-y\)

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

1. 1+ x² + 2x- y²

= (x² + 2x+1) -y²

= (x+1)² -y²

= (x+1-y)(x+1+y) 

2. x²+ x + y² + y + 2xy

= (x² + 2xy + y²) + (x+y)

= (x+y)² + (x+y)

=(x+y)(x+y+1) 

1. x² + x - y² +y 

= (x²  -y²) + (x+y)

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

= (x+y)(x-y+1)

2. 4x² - 9y² + 4x -6y

= (2x)² -(3y)² + 2(2x - 3y)

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

= (2x-3y)(2x+3y+2) 

3. x² - 9 - 5x - 15 

= x² - 5x - 24

= x² - 8x + 3x -24

= x(x-8) + 3(x-8)

= (x-8)(x+3)

b, (1−7x)(2x−3)−(14x−9)(5−x)=30

2x−14x^2−3+21x−(70x−45−14x^2+9x)=30

−56x+42=30

−56x=−12

x=3/14

8ax - 4ay - 2x + y

= (8ax - 4ay) - (2x - y)

= 4a(2x - y) - (2x - y)

= (2x - y)(4a - 1)

8ax-4ay-2x+y

=4a(2x-y)-(2x-y)

=(4a-1)(2x-y)

a) a^2+b^2+c^2=ab+ac+bc. cm a=b=c

Ta có: a^2 + b^2 + c^2 = ab + bc + ca 

<=> 2.a^2 + 2.b^2 + 2.c^2 = 2.ab + 2.bc + 2.ca 

<=> ( 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 (1) 

Vì (a-b)^2 ; (b-c)^2 ; (c -a)^2 ≧ 0 với mọi a,b,c. 

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

Từ (1) và (2) khẳng định dấu "=" khi: 

a - b = 0; b - c = 0 ; c - a = 0 => a=b=c 

Vậy a=b=c  ( đpcm )

b)a+b+c=o cm a^3+b^3+c^3=3abc

thay a^3+b^3=(a+b)^3 -3ab(a+b) .Ta có :

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+c)=0

<=>(a+b+c). [(a+b)^2 -c.(a+b)+c^2] -3ab(a+b+c)=0

<=>(a+b+c).(a^2+2ab+b^2-ca-cb+c^2-3ab)...

<=>(a+b+c).(a^2+b^2+c^2-ab-bc-ca)=0

luôn đúng do a+b+c=0

c)tìm x 8x^3-12x^2+6x=28

8x^3 - 12x^2 + 6x - 28 = 0

8x^2(x-2)+4x(x-2)+14(x-2)=0

<=>2(x-2)(4x^2+2x+7) = 0

Ta có 4x^2+2x+7=(2x)^2+2.2x1/2 +1/4 -1/4+28/4=(2x+1/2)^2+27/4 >0 V x

=>x-2=0 <=>x=2

Vậy PT có tập ngiệm là  S ={2}

\(5\left(x-2\right)+2>1-2\left(x+1\right)\)

\(\Leftrightarrow5x-10+2>1-2x-2\)

\(\Leftrightarrow7x>7\)

\(\Leftrightarrow x=1\)

\(5+3x\left(x+3\right)< \left(3x-1\right)\left(x+2\right)\)

\(\Leftrightarrow5+3x^2+9x< 3x^2-x+6x-2\)

\(\Leftrightarrow4x< -7\)

\(\Leftrightarrow x< \frac{-7}{4}\)

\(\frac{x+3}{3x-2}>1\)

\(x+3>3x-2\)
 

\(2x< 5\)

\(x< \frac{5}{2}\)