Let-z-1-and-z-2-be-two-roots-of-the-equation-z-2-az-b-0-z-being-complex-Further-assume-that-the-origin-z-1-and-z-2-form-an-equilateral-triangle-Then-

Question Number 25075 by Mr easy last updated on 03/Dec/17

Let z_{1} and z_{2} be two roots of the equation

z^{2} + a z + b = 0, z being complex . Further

assume that the origin, z_{1} and z_{2} form

an equilateral triangle . Then,

Answered by ajfour last updated on 03/Dec/17

z_{2} = z_{1} e^{\pm i \frac{π}{3}} a n d z_{1} z_{2} = b

z_{1} + z_{2} = - a

\Rightarrow z_{2}^{3} = - z_{1}^{3}

o r (z_{1} + z_{2}) [{(z_{1} + z_{2})}^{2} - 3 z_{1} z_{2}] = 0

\Rightarrow - a [a^{2} - 3 b] = 0

\Rightarrow a = 0 o r a^{2} = 3 b .

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