If the roots of the equation are equal, prove that either a = 0 or a³ + b³ + c³ = 3abc.

Question

If the roots of the equation are equal, prove that either a = 0 or a 3 + b 3 + c 3 = 3abc.

Philoid · Accepted Answer

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

D = b² – 4ac

If D = 0, roots are equal

Given, roots of are equal.

∴ D = 0

⇒ 4(a² – bc)² – 4(c² – ab)(b² – ac) = 0

⇒ a⁴ + b²c² – 2a²bc – b²c² – a²bc + ab³ + ac³ = 0

⇒ a(a³ + b³ + c³ – 3abc) = 0

⇒ a = 0 or a³ + b³ + c³ = 3abc

If the roots of the equation are equal, prove that either a = 0 or a3 + b3 + c3 = 3abc.