Is there a technology that we can make negative energy?

Energy is neither positive or negative. When matter is considered as container of energy, then energy is always conserved and cannot be created or destroyed.

The big bang theory does not state that the universe started from nothing. The big bang states that the universe was created from a singularity that contained all the ingredients to make the universe.

"Is there a technology that we can make negative energy?"

No. By definition, all energy is "positive", it all does work.

"If positive energy is created then negative energy is also created because energy cannot be created or destroyed."

Negative energy is meaningless. Energy is not even defined in an expanding Universe.

"At the start of the universe there is nothing,"

Maybe. It is expected what we started with pure (positive) energy.

"but the big bang created everything,"

Things, yes, but not energy.

"there is equal quantities of negative and positive energies."

False. You are thinking there is an "anti"energy analog to matter and antimatter. There isn't.

"Can we make negative energy on earth?"

No.

"Update 1: So we can just send these negative energies into space and use positive energy for our own benefit, e.g. electricity as positive energy."

No.

No - although it is possible to use the 'Casmir Effect' to create the illusion of negative energy.

Junk food and video games. Ask any lazy fat kid.

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