If the roots of the equation (c2 - ab)x2 - 2(a2 - bc) x + b2 - ac = 0�

1.	If the roots of the equation (c2 - ab)x2 - 2(a2 - bc) x + b2 - ac = 0 are equal, prove that either a = 0 or a3 + b3 + c3 = 3abc.
Answer» For a quadratic equation, ax² + bx + c = 0, D = b² – 4ac If D = 0, roots are equal Given, roots of (c² - ab)x² - 2(a² - bc) x + b² - ac = 0 are equal. ∴ D = 0 ⇒ 2(a² – bc)]² – 4(c² – ab)(b²– ac) = 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 (c2 - ab)x2 - 2(a2 - bc) x + b2 - ac = 0 are equal, prove that either a = 0 or a3 + b3 + c3 = 3abc.

Answer»

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

D = b² – 4ac

If D = 0, roots are equal

Given,

roots of (c² - ab)x² - 2(a² - bc) x + b² - ac = 0 are equal.

∴ D = 0

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