Take the difference to give the width of the tunnel at this height:
9.06 - 2.94 = 6.12 m
So yes, the truck can pass underneath because it has a width 3.5 m at a height above the tunnel floor of 5 m where as the tunnel has a width of 6.12 m at this height above the floor (y=0).
So in answer to your question: "Isn't the bridge only 3.5 m wide underneath?"
No, it is the truck that is 3.5 m wide.
Interesting that your teacher didn't recognise that the solution to the problem wasn't simply the solution to the quadratic but the difference between the solutions!
I hope you understand this better now. If not I urge you to sketch a picture of the quadratic along with the x and y axes and the horizontal line y=5.
Note: Just a little advice; it appears your maths is really good but I know how easy it is, when getting lost in reams of equations, to loose sight of the problem you are trying to solve. It is a really good idea to keep clear in your mind what the solution is you are looking for. I'm not moaning here, really, it is a mistake I STILL make and I've done a lot of mathematical problem solving in my life.
Anyway happy mathematic-ing!