Re: x²+4x+5=0

On 1/22/2025 2:57 PM, Chris M. Thomasson wrote:

On 1/22/2025 2:42 PM, Moebius wrote:

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

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

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

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It has the two "roots" (solutions):
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x = -2 - i
x = -2 + i
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Hint:
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x² + 4x + 5 = (x + 2)² + 1.
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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.
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The x^2 component has two roots.

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Doesn't make much sense. :-P

 Shit happens. When I see x^2 and think of complex numbers, I think of two roots. This can imply that there are two solutions.

For instance, this should have three roots?
x^3+1+x^2 = 0
What am I missing here? Sorry if it's something obvious.
Shit.