Question

if x^2+y^2+z^2-xy-yz-zx=0,prove that x=y=z

Answer 1

x² + y² + z² -xy - xz - yz = 0
multiplying 2 on both sides

2x² + 2y² + 2z² - 2xy - 2xz - 2yz = 0
x² + y² - 2xy+ x² + z² - 2xz + z² + y² - 2yz = 0
(x - y)² + (x - z)² + (y - z)² = 0

so the value are in square so they can't be negative so they must be individually zero to satisfy the equation.

So, x - y = 0 ⇒ x = y
x - z = 0 ⇒ x = z
y - z = 0 ⇒ y = z

Hence, x = y = z