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Are there any specific rules for proving theorems ? For example when we solve quadratic equations we can prove any quadratic equation by factoring or completing the square or by formula . I mean are there any specific steps that we can use to prove any theorem ? or we have to just memorize previously proved theorems and postulates to prove new theorems like we do in algebra when we factorise polynomials we learn to recognize patterns and memorize special identities .

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    What? Hell no. There is no general rule to prove anything. Otherwise there would be no unproved (or disproved) statements. Your experience and intuition might help you but they definitely don't guarantee achieving results. That being sad there are certain tools that can help in certain classes of problems. – 2017-02-20
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    Memorizing other theorems definitely isn't the way to go. It's necessary to thoroughly understand why certain proofs work the way they do so that you can build up an intuition as to what will work in various scenarios. Not a trivial task and I'd go so far as to say it's not a task that anyone ever really "finishes." – 2017-02-20
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    Proving theorems is an art and can (luckily) not be automated. Then all mathematicians would be unemployed. So _No_, there are no rules or guidelines. – 2017-02-20
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    I wouldn't call solving a quadratic a theorem but a *method*,are you sure you wanted to ask about theorem proving or rather you wanted to ask how to learn/apply identities/methods? – 2017-02-20

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The truth may be hilarious for you but the only rule is that each step in proof should be faithful and rigorous conclusion from the previous ones.

It isn't possible to provide general algorithm for proof-creation at least due to the fact that there are some statements that we can't prove at all. If you are curious enough read more about Godel's incompleteness theorems.

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Sadly there is no universal method for proving conjectures. If there was we surely would have proven (or disproven) everything we could come up with by now. There are some common strategies such as "proof by induction" or "reducto ad absurdum", but these methods don't always help and they certainly aren't the only ways to prove conjectures.

The only true advice I can give to someone trying to get better at proving conjectures is to study as much as you can about what already has been proven so you don't have to reinvent the wheel to implement your own proof.