Question

The roots z1,z2,z3 of the equation x^3+3ax^2+3bx+c=0, who triangle will be an equilateral triangle if a^2=b

Answer 1

Answer:

Step-by-step explanation:

If z₁, z₂ and z₃ are vertices of an equilateral triangle, then we know that

z₁² + z₂² + z₃² = z₁z₂ + z₂z₃ + z₃z₁.--------(*)

Now, given z₁, z₂ and z₃ re the roots of the equation

x³+3ax²+3bx+c = 0 ,

Hence sum of its root ,  z₁+z₂+z₃ = -3a,----(1)

sum of its root taken 2 at a time , z₁z₂ + z₂z₃ + z₃z₁ = 3b.----(2)

Now, consider (z₁+z₂+z₃)² = z₁² + z₂² + z₃² + 2z₁z₂ +2 z₂z₃ + 2z₃z₁.

=> 9a² = 3(3b)

=> a² = b.