I can't help you with 1 and 2, I hate those kind of questions.

For 3, take the derivative of that function, find the critical points, and then you can show that 1 is the only time the function touch the x axis.

For 4, you know one root is , the other solution is , substitute these values to x, you get 2 equations, solve the simultaneous equations, and you'll get your b and c value.

Another way of doing it is with the root coefficient relationship. In the quadratic equation , suppose the roots are and ,