Question

Prove that :  x²+ y² + z² - xy - yz - zx  is always positive.

Answer 1

x² + y² + z² - xy - yz - zx

= 2(x² + y² + z² - xy - yz - zx)

= 2x² + 2y² + 2z² - 2xy - 2yz - 2zx

= x² + x² + y² + y² + z² + z² - 2xy - 2yz - 2zx

= (x² + y² - 2xy) + (z² + x² - 2zx) + (y² + z² - 2yz)

= (x - y)² + (z - x)² + (y - z)²

Since square of any number is positive, the given equation is always positive.