Problem
ALG-B2-M06-P004 Differences and p2 minus 3q
#4
★★☆☆☆ Level 2 of 5
Prove \[p^2-3q=\frac12\left((a-b)^2+(b-c)^2+(c-a)^2\right).\]
Hint. Expand the right side and compare it with \(p^2-3q\).
The right side equals \(\frac12(2a^2+2b^2+2c^2-2ab-2bc-2ca)=a^2+b^2+c^2-q\). Also \(p^2-3q=(a^2+b^2+c^2+2q)-3q=a^2+b^2+c^2-q\).
Gives a quick way to see \(p^2\ge3q\).