Is there any difference between "solved the problem" and "fixed the problem"? Do people actually use both forms?
I would suggest 'to solve' is to do with understanding, while 'to fix' is to do with implementing.
For example, I have been attempting to solve (understand the cause & possible solutions for) a [medical | computer | mechanical |...] problem. Once I have solved the problem, I may be able to implement a fix for the problem using the potential solution/s I have identified. Or, I may not.
Generally speaking there are two slight and relatively subtle differences.
One, solving the problem means identifying how the problem is solved.
If the problem is a knowledge problem, or an abstract issue, this is all that is required.
Some problems exist in concrete reality. To fix a concrete problem, the abstract solution also has to be made concrete. Fixing the problem in those cases means carrying out the actions that provide the solution.
As an example: my dishwasher has stopped working.
This is a concrete problem.
To solve the problem, you could identify the fact that the filter has been clogged up, and that cleaning the filter would fix it. The problem is solved, but not yet fixed.
Performing the concrete action of cleaning out the filter will fix the problem.
The use of synecdoche means that solve can be used for situations where you both solve and fix a problem. And fix can be used for situations where you both solve and fix a problem.
In cases where the problem is purely abstract - for example, a problem in maths - the problem would be solved, not fixed.
There is one good example. You sometimes hear people say "Can you fix me (up)?" to a doctor, but you don't hear "Can you solve me?"
If there is a body or concrete things that are broken (figuratively for a body), you fix it. You never solve it. You never solve a broken machine, car, watch, etc.
Another example is, you solve the crime, not fix the crime as it takes complicated steps, i.e. you need physical evidence, statements from eyewitnesses, your own analysis report, etc.
"To solve" is used for something that is not easy without taking complicated measures or thinking process. "To fix" is used for something that takes mechanical (surgical for a body) procedures. You can fix a car, but you can't fix a mystery or a math problem.
However, there is a borderline. One thing to note is politicians seem to like the word fix and they use it for the economy. However, if you use the "economic problem", you use solve more broadly than fix. The below Ngram Viewer shows it.
Edit: You use solve for X problem because it is more idiomatic as the word problem fits better with solve. You can fix the economy, but you rarely use fix for the economic problem, however, you can solve the economic problem. You don't solve the economy.
You don't solve the TV set. You fix the TV set. You solve the TV set Problem.
You don't solve the computer. You fix the computer. You solve the computer problem.
Solving is finding the solution to a logical problem by analyzing it. Fixing is making a physical change to something to reach a convenient state.
However, the fix could be a result of solving the problem as you apply the output of your solving (the solution) to fix something.
You solve a math problem but you do not fix it.
You fix a broken window but you do not solve it.
I hope I could deliver the idea clearly.
"Solved" implies that you understood the problem (and depending on the context, usually also that you made it go away). So you solved it the way you'd solve a math problem.
"Fixed" might mean you didn't understand the problem, but got it to go away. e.g. you changed something in a computer program, and then it started doing what you want, but you don't understand why. That's fixing without solving. Usually it's just another way of saying "solved the problem", with the emphasis on "it's working now". Usually there's no implication that you didn't understand the fix.
If someone says "I fixed the problem", it usually means they think they've made a permanent fix that won't break again. Depending on context and who you're talking to (their level of expertise with the thing they fixed), they might or might not have understood exactly why their actions solved the problem. (And in that last sentence, "solved" means "removed" or "corrected", not necessarily with the connotation of fully understood. Context matters as much as word choice, in some cases.)
One crude example of a temporary fix (which most people know is only a temporary fix) is to bang on something (like an old TV set) until it works. Banging on something can make a loose connection temporarily connect, but it will probably come loose again soon. You might describe this as "fixing the problem", but not as "solving the problem".
Also note that "fix" can apply to something that has a problem, but "solve" can only apply to the problem itself.
- good: "I fixed the TV".
- good (but overly verbose): "I fixed the problem with the TV".
- good: "I solved the problem with the TV."
- bad: "I solved the TV."
Suggestion: If you're going to say anything about how you solved the problem, or what the problem was, a long sentence with solve makes sense. There's nothing at all wrong or weird-sounding with still using "fix" in this case, though.
Otherwise it's easier to just say you fixed the thing, rather than the problem with the thing. Unless there were multiple problems and you only fixed one: fixing a thing implies that the thing is now working.
"Solved" implies that you had to figure out what was wrong before you could "fix" the problem.
If your car has a flat tire, you fix the problem by replacing the flat tire with a good one.
If your car won't start, you could, for example, solve the problem by checking the spark plugs and noticing that they are worn out. You would then fix the problem by replacing the spark plugs.
I think that "fixed" involves a repair of some kind while "solved" is focused more on discovering/applying a full answer in some form. Where those two concepts overlap the words might be interchangeable.
Many fixes are quick and temporary and don't get to the root of a problem or they don't involves any kind of creative ingenuity so we don't always think of them as solutions.
And when talking about solving math problem, for example, it wouldn't make sense to refer to "fixing" them since they were never "broken". At the same time you might find a mistake in your work which needed fixing.
'Solved' means that the problem no longer exists.
'Fixed' means that you applied something to overcome the issue.
With 'solved' - the problem can have gone away entirely.
With 'fixed' the situation may be ok now - but it could be that your repair, is still glued on the side of it, which is continuing to actively need to be there, fixing the problem.
Sometimes 'solved' can also mean that your repair is still glued on the side - but 'solved' doesn't necessarily express that.