If the roots of the equation 
are equal, then prove that 2b = a + c.
For a quadratic equation, ax2 + bx + c = 0,
D = b2 – 4ac
If D = 0, roots are equal
⇒ (c – a)2 – 4(b – c)(a – b) = 0
⇒ c2 + a2 – 2ac + 4b2 – 4ab - 4cb + 4ac = 0
⇒ a2 + 4b2 + c2 + 2ac – 4ab – 4bc = 0
⇒ (a – 2b + c)2 = 0
⇒ 2b = a + c
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