Let a,b, and c be nonzero real numbers, and let x=cb+bc,y=ca+ac,z=ba+ab.Simplify x2+y2+z2−xyz.
Solution — click to reveal
Substituting and expanding, we get x2+y2+z2−xyz=(cb+bc)2+(ca+ac)2+(ba+ab)2−(cb+bc)(ca+ac)(ba+ab)=c2b2+2+b2c2+c2a2+2+a2c2+b2a2+2+a2b2−(c2a2+c2b2+1+a2b2+b2a2+1+b2c2+a2c2)=4.