For a quadratic equation, ax² + bx + c = 0,

D = b² – 4ac

If D > 0, roots are real.

⇒ x² – (a + b)x + ab + x² – (b + c)x + bc + x² – (a + c)x + ac = 0

⇒ 3x² - 2(a + b + c)x + ab + bc + ac = 0

⇒ D = 4(a + b + c)² – 12(ab + bc + ac)

⇒ D = a² + b² + c² + 2ab + 2ac + 2bc – 3ab – 3bc – 3ac

⇒ D = 1/2 × (2a² + 2b² + 2c² - 2ab – 2ac – 2bc)

⇒ D = 1/2 × ((a – b)² + (b – c)² + (c – a)²)

Thus, D is always greater than 0, and the roots are real

Now, when a = b = c,

D = 0, thus the roots are equal when a = b = c.