Re: x²+4x+5=0

Am 22.01.2025 um 23:16 schrieb Chris M. Thomasson:

On 1/22/2025 5:48 AM, Richard Hachel wrote:

x² + 4x + 5 = 0
>
This equation has no root, and <bla>

It has the two "roots" (solutions):
x = -2 - i
x = -2 + i
Hint:
x² + 4x + 5 = (x + 2)² + 1.
Hence x² + 4x + 5 = 0 is equivalent with (x + 2)² + 1 = 0 or (x + 2)² = -1. So x + 2 has to be a (one or more) complex number(s) z such that z² = -1. We know such numbers, they are i and -i (and there aren't more). Hence we have x + 2 = i or x + 2 = -i resp. And hence x = -2 + i or x = -2 - i resp.

The x^2 component has two roots.

Doesn't make much sense. :-P