Question

[nguyen thi thu trang]

c/m các đẳng thức sau:

a)(x+y+z)²-(x-y-z)²=4x(y+z)

b)(2a+b)²-(a+b)²-3a²=2ab

Answer 1

[Võ Đông Anh Tuấn]

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

\(=\left(x+y+z-x+y+z\right)\left(x+y+z+x-y-z\right)\)

\(=4x\left(y+z\right)=VP\)

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

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

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

\(=3a^2+2ab-3a^2=2ab=VP\)

Answer 2

[Nguyễn Anh Duy]

a) \(\left(x+y+z\right)^2-\left(x-y-z\right)^2=4x\left(y+z\right)\)

\(\Rightarrow x^2+y^2+z^2+2xy+2xz+2yz-\left(x^2+y^2+z^2-2xy-2xz+2yz\right)=4x\left(y+z\right)\)\(\Rightarrow x^2+y^2+z^2+2xy+2xz+2yz-x^2-y^2-z^2+2xy+2xz-2yz=4x\left(y+z\right)\)\(\Leftrightarrow4xy+4xz=4x\left(y+z\right)\)

\(\Leftrightarrow4x\left(y+z\right)=4x\left(y+z\right)\).

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

\(\Rightarrow\left(2a\right)^2+2.2a.b+b^2-\left(a^2+2ab+b^2\right)-3a^2=2ab\)

\(\Rightarrow4a^2+4ab+b^2-a^2-2ab-b^2-3a^2=2ab\)

\(\Leftrightarrow2ab=2ab\)