\(\left(a+b-c\right)^2-\left(a-c\right)^2-2ab+2bc\)

\(=a^2+b^2+c^2+2ab-2ac-2bc-a^2+2ac-c^2-2ab+2bc\)

\(=b^2\)

ko biết

khai triển các vế ra rồi thu gọn

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

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

\(b,\)\(\left(2x-3\right)\left(4x^2+6x+9\right)-\left(54+8x\right)\)

\(=8x^2-27-54-8x=8x^2-8x-81\)

\(c,\)\(\left(3x+y\right)\left(9x^2-3xy+y^2\right)-\left(3x-y\right)\left(9x^2+3xy+y^2\right)\)

\(=27x^3+y^3-\left(27x^3-y^3\right)=2y^3\)

\(d,\)\(\left(a+b+c\right)^2-\left(a-c\right)^2-2ab+2bc\)

\(=a^2+b^2+c^2+2ab+2bc+2ac-a^2+2ac-c^2-2ab+2bc\)

\(=b^2+4bc+4ac\)

Dùng hằng đẳng thức A² + 2AB + B² = ( A + B)² :

Ta được ... = (a-b+c+b-c)² = a²

 (a+b-c)^2 + (a-b+c)^2 - 2(b-c)^2 

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

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

= a^2 + a^2 

= 2.a^2

Ta có  : (a + b - c)² + (a - b + c)² - 2(b - c)²

= a² + b² + c² + 2ab - 2bc - 2ca + a² + b² + c² + 2ca - 2ab - 2bc - 2(b² - 2bc + c²)

= a² + b² + c² + 2ab - 2bc - 2ca + a² + b² + c² + 2ca - 2ab - 2bc - b² + 2bc - c²

= 2a² + b² + c² - 2bc 

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

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

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

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

= 2[(a + b)^2 + (a - b)^2] + 4c^2 

=2(2a^2 + 2b^2) + 4c^2 

= 4(a^2 + b^2 + c^2)

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

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

= 0 + ( b-b ) + ( c+c )

=0+0+  2x

=2x

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

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

=-a-b+c+a+b+c

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

=0+0+c.2

=c2