Simplify the Following Expression
\[ (x+y+z)^2+\left(x+\frac{y}{2}+\frac{z}{3}\right)^2-\left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^2 \]
Solution:
Using identity:
\[ (a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca \]
\[ (x+y+z)^2 = x^2+y^2+z^2+2xy+2yz+2zx \]
\[ \left(x+\frac{y}{2}+\frac{z}{3}\right)^2 = x^2+\frac{y^2}{4}+\frac{z^2}{9} +xy+\frac{yz}{3}+\frac{2xz}{3} \]
\[ \left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^2 = \frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16} +\frac{xy}{3}+\frac{yz}{6}+\frac{xz}{4} \]
\[ (x+y+z)^2+\left(x+\frac{y}{2}+\frac{z}{3}\right)^2-\left(\frac{x}{2}+\frac{y}{3}+\frac{z}{4}\right)^2 \]
\[ = \frac{7x^2}{4} +\frac{41y^2}{36} +\frac{127z^2}{144} +\frac{8xy}{3} +\frac{13yz}{6} +\frac{29xz}{12} \]