conventionsopticsreflection
According to the Cartesian sign convention, the focal length should be positive since the convex mirror forms a virtual image to the right of mirror (positive direction) when light comes from left.
My teacher also said that it would be positive but I have read in many places on the internet that it is negative. How?
The "rule" that real images are always inverted is not correct. That rule might work when you have only a single optical element (like a lens), but not necessarily when you have two or more.
Take a look at this:
That's a real, upright (aka erect) image labeled $I_B$. You can tell it's real because the rays at the final image actually converge at that physical location, unlike virtual images whose location of "convergence" does not actually have physical rays passing through it (only the "backtracking" rays one typically draws).
As for your projector and mirror, you can draw a ray diagram carefully (if you know the internal workings of a projector) and apply the same test to see if the image is virtual or real. But do remember that projectors can be modified so that the image you see is inverted to accommodate mirrors and such. Anyway, to get something projected on to a screen, I do believe the image needs to be real, so I would say it is indeed a real image you're seeing.
Finally, here's a simple example of two elements that again goes against the rules in your book:
The situation above has a real, inverted image using a lens and mirror (a mirror!). Thus, the rules you've been given aren't general enough to deal with multiple-element situations.
This plano concave lens has the same object and image positions as your silvered plano convex lens. Hence the same focal length. The sign convention comes from the sign convention of the u and f distances.
Best Answer
Every time you look up "the" spherical mirror formula, it comes with a set of "where's". These define what each symbol stands for, and the sign convention to use to distinguish the location of objects and images and the difference between concave and convex radii.
You can find different-looking spherical mirror formulas, with (naturally) different sets of "where's". These can each be applied to a specific problem and give a different-looking answer, which is interpreted by the "where's" to give the same result.
You can get in a lot of trouble by combining one version of the formula with a some other version of "where's"...