Complex Numbers 2

Algebra Level 4

If $\alpha$ is complex and $x^2+\alpha x+\overline{\alpha}=0$ has a real root $\lambda$ , then $\lambda$ is equal to

2\left(\alpha+\overline{\alpha}\right)

None of the others

\alpha+\overline{\alpha}

1 solution

Sherwin D'souza
Apr 29, 2016

The equation is given as

$x^2+αx+\overline{α}=0$

If we take conjugate on both sides,

$x^2+\overline{α}x+α=0$

We subtract both equations.

$x^2+\overline{α}x+α=0$

$x(α-\overline{α})+(\overline{α}-α)=0$

$x(α-\overline{α})=α-\overline{α}$

$x=\frac{α-\overline{α}}{α-\overline{α}}$

Hence x=1