Question

If x, y and z are three different numbers, then prove that: 2 + y2 + z -xy - yz - zx is always positive​

Answer 1

Correct question :

• If x, y and z are three different numbers, then prove that: x² + y² + z² - xy - yz - zx is always positive

Solution :

= x² + y² + z²- xy - yz - zx

= 2x² +2y² +2z²-2xy-2yz-2zx/2

(Multiplying and dividing by 2)

= x²+y²-2xy+y² +z²-2yz +z² +x²-2zx/2

= ½[(x - y)² + (y - z)²+ (z - x)² ]

(x - y)² + (y - z) ² + (z - x)² is always positive as they are the sum of squares.

∴ x² + y² + z²- xy - yz - zx is always positive.