MathQuill-Evaluate builds on MathQuill, and gives you basic capabilities for evaluating the expressions that users enter into your MathFields. Just pass your MathField as an argument to the evaluate function, and get a solution back.
MathQuill-Evaluate does this by passing the LaTeX from your MathField to Wolfram|Alpha. Wolfram|Alpha has the ability to evaluate arbitrary LaTeX and return either a simplified expression or a solution.
The Wolfram|Alpha API is just awful, but I managed to make it cooperate a little bit. However, this isn't the stablest of projects right now. If you're trying to evaluate anything seriously, I'd recommend building your own, more robust solution, but this will work in many trivial cases.
This is a live example. This example displays LaTeX as you type, and then when you press "Evaluate", that LaTeX is sent to Wolfram|Alpha. After a few seconds, you should see a response.