1) Why are there usually two solutions in quadratic equations?
I think it'd be a useful for you to look at your choice of words and ask;
Why usually and not always? What are solutions? Can a quadratic equation have no solutions?
What is a quadratic equation? Why equation and not expression? Is there a difference?
Is x^2 - 2x = 3x - 6 a quadratic equation? If so why?
Is x(x+2) = -4 a quadratic.
If I said any quadratic equation can be written as ax^2 + bx + c = 0 is this true? What would the above equations look like?
Think about factorising a^2 + bx + c into (cx + d)(ex + f).
Even if it was true why bother with the 0? Hint (think about the values of a and b that make a x b = 0)
Draw graphs of the equations that you know the solutions for. Does the shape of the graph suggest a reason why your statement is or isn't true?
Please note that these are just my SUGGESTIONS and are only one way to help you answer the question.
2) I'm not sure exactly what you mean but consider
x^2 - 5x + 6
___________ = (x - 3)
(x - 2)
It is true that (x - 2)(x - 3) = x^2 - 5x + 6 so what's the problem? Hint: look at different values for x such as 1, 2, 3. Does the right hand side always equal the left?
Hope this has been of some help. I've just qualified as a teacher and am trying to get some practice in before september!