Solutions?

Algebra Level 4

How many solutions for $x$ exist satisfying $3^{2x}+(2\times 3^{x})+1=0$ ?

Note: solutions may be real or complex.

Infinitely many 1 2 0

1 solution

Miles Koumouris
Apr 14, 2017

Let $3^{x}=a$ ; then

$a^{2}+2a+1=0\Longrightarrow (a+1)^{2}=0\Longrightarrow a=-1$ .

So $3^{x}=-1\Longrightarrow$ no real solutions exist.

But now let $3^{x}=-1=e^{i(2k+1)\pi }, k\in \mathbb{Z}$ ;

then $\textrm{ln}(3^{x})=i(2k+1)\pi \Longrightarrow x=\dfrac{i(2k+1)\pi }{\textrm{ln}(3)}, k\in \mathbb{Z}$ .

So infinitely many solutions exist.

Damnit, I included only the case where k = 0.

Konstantin Zeis - 4 years, 1 month ago