Solve 2(y^2+z^2+x^2-yz-yx-zx)+(x+y+z)^2

$$2 ( y ^ { 2 } + z ^ { 2 } + x ^ { 2 } - y z - y x - z x ) + ( x + y + z ) ^ { 2 }$$

Evaluate

$$3\left(x^{2}+y^{2}+z^{2}\right)$$

Solution Steps

Use the distributive property to multiply 2 by y^{2}+z^{2}+x^{2}-yz-yx-zx.

$$2y^{2}+2z^{2}+2x^{2}-2yz-2yx-2zx+\left(x+y+z\right)^{2}$$

Square x+y+z.

$$2y^{2}+2z^{2}+2x^{2}-2yz-2yx-2zx+x^{2}+2xy+2xz+y^{2}+2yz+z^{2}$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$2y^{2}+2z^{2}+3x^{2}-2yz-2yx-2zx+2xy+2xz+y^{2}+2yz+z^{2}$$

Combine -2yx and 2xy to get 0.

$$2y^{2}+2z^{2}+3x^{2}-2yz-2zx+2xz+y^{2}+2yz+z^{2}$$

Combine -2zx and 2xz to get 0.

$$2y^{2}+2z^{2}+3x^{2}-2yz+y^{2}+2yz+z^{2}$$

Combine 2y^{2} and y^{2} to get 3y^{2}.

$$3y^{2}+2z^{2}+3x^{2}-2yz+2yz+z^{2}$$

Combine -2yz and 2yz to get 0.

$$3y^{2}+2z^{2}+3x^{2}+z^{2}$$

Combine 2z^{2} and z^{2} to get 3z^{2}.

$$3y^{2}+3z^{2}+3x^{2}$$

Expand

$$3x^{2}+3y^{2}+3z^{2}$$

Solution Steps

Use the distributive property to multiply 2 by y^{2}+z^{2}+x^{2}-yz-yx-zx.

$$2y^{2}+2z^{2}+2x^{2}-2yz-2yx-2zx+\left(x+y+z\right)^{2}$$

Square x+y+z.

$$2y^{2}+2z^{2}+2x^{2}-2yz-2yx-2zx+x^{2}+2xy+2xz+y^{2}+2yz+z^{2}$$

Combine 2x^{2} and x^{2} to get 3x^{2}.

$$2y^{2}+2z^{2}+3x^{2}-2yz-2yx-2zx+2xy+2xz+y^{2}+2yz+z^{2}$$

Combine -2yx and 2xy to get 0.

$$2y^{2}+2z^{2}+3x^{2}-2yz-2zx+2xz+y^{2}+2yz+z^{2}$$

Combine -2zx and 2xz to get 0.

$$2y^{2}+2z^{2}+3x^{2}-2yz+y^{2}+2yz+z^{2}$$

Combine 2y^{2} and y^{2} to get 3y^{2}.

$$3y^{2}+2z^{2}+3x^{2}-2yz+2yz+z^{2}$$

Combine -2yz and 2yz to get 0.

$$3y^{2}+2z^{2}+3x^{2}+z^{2}$$

Combine 2z^{2} and z^{2} to get 3z^{2}.

$$3y^{2}+3z^{2}+3x^{2}$$