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A teacher gave us a challenge to solve an equation , we spent hours and hours trying to solve the equation without any result , and in a late time my friend called me and told me that he solved the equation , I said to him :

“how could you have solved this equation? “

Is this sentence could be used to express astonishment ?

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That isn't correct.

You use "could have" to indicate surprise at someone's ethics or morality.

How could you have cheated on your girlfriend!?

You could express surprise and admiration in lots of ways. It is simple to be explicit:

Wow that's great!

And then ask "How"

How did you solve it?

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    If you say "How could you have solved the equation?", you are saying the other person is too stupid to solve the equation. – Michael Harvey Jul 27 '19 at 20:52
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    There are other opinions. – Michael Login Jul 27 '19 at 20:53
  • @MichaelHarvey not necessarily. It could be that no one has been able to solve it before, e.g. Fermi's Conjecture. The expression indicates surprise and possibly disbelief, but any negative implication depends on context. – Andrew Jul 28 '19 at 1:44

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