Is it technically incorrect to write proofs forward? A question on an assignment was similar to prove:
$$2a^2-7ab+2b^2 \geq -3ab.$$
and my proof was:
$$2a^2-4ab+2b^2\geq0$$
$$a^2-2ab+b^2\geq0$$
$$(a-b)^2\geq0$$
which is true.

However, my professor marked this as incorrect and the "correct" way to do it was:

Starting from $$(a-b)^2\geq0$$ we have:
$$a^2-2ab+b^2\geq0$$
$$2a^2-4ab+2b^2\geq0$$
$$2a^2-7ab+2b^2 \geq -3ab.$$

His point was that if we start with a false statement we can also reduce it to a true statement (like $-5 =5$ we can square for $25 = 25$). I argued however that you can go back and forth between my operations (if and only if, which doesn't work for the $-5 =5$ example). He still didn't give me the marks for it. Which leads me to my questions:


Is my proof equally valid?
Do real mathematicians all write one way or the other when writing in a paper?
There are two ways to interpret your argument (which hardly counts as a proof if this is exactly what you have written in your assignment).

First Way
You actually meant to write, \begin{align}2a^2-7ab+2b^2 \geq -3ab&amp;\implies 2a^2-4ab+2b^2\geq0\\&amp;\implies a^2-2ab+b^2\ge 0\\&amp;\implies (a-b)^2\ge0\end{align}
Second Way
You actually meant to write, \begin{align}2a^2-7ab+2b^2 \geq -3ab&amp;\iff 2a^2-4ab+2b^2\geq0\\&amp;\iff a^2-2ab+b^2\ge 0\\&amp;\iff (a-b)^2\ge0\end{align}

In the second case your argument is correct but in the first case it is not (as your professor has elaborated via an example).
Now the answer to your questions,

If you didn't say explicitly in your assignment in which way your argument is to be interpreted and then if your teacher interprets your argument in the first way, you can't blame him/her for not giving you the corresponding marks of the question simply because it was you who failed to be explicit )and the first way of interpreting your argument indeed shows a misunderstanding of the working of $\implies$). So, if I were in the position of your professor, I would give you no marks.

I don't know what you mean by &quot;one way or the other&quot; here. The best way to answer this question will be to read the papers of some real mathematicians. However, when real mathematicians write a paper it is in general clear what they are assuming to be true and how the steps lead to the conclusion (which is not the case in your argument as I have explained above).
The teacher is correct, but hasn't explained it well.

Working one way doesn't always prove the opposite way. This is less obvious in basic maths, but it's important to grasp.  Why it doesn't becomes more obvious when you consider the kinds of operations used. With operations like inequalities, complex numbers, set inclusion and undefined operations like division (by zero), it is often the case that what works one way just doesn't prove anything about what works the other way, unless you actually check it will work the other way. But you have to specifically check that.

Whichever way the answer comes to mind, you will have to work it out the right way to actually prove anything (rather than just suspect it)