Let's use brute force... for the first question :
If are the roots, then and
Let be the equation which roots are
So the equation you're looking for is
How do i solve these?
1) If and are the roots of the equation = 0, then the equation whose roots are and , is?
2) If the equations and have a common root, then ?
3) If the difference of the roots of the equations be 1, then:
For question 2)...
There exists such that :
You may be able to solve for by yourself eh ?
For question 3)...
We know that the sum of the two roots is -b and their product is c.
So we have
My approach to #1 is similar to Moo's . . . with a different answer.
. . Did I mess up?
1) If and are the roots of the equation: ,
find the equation whose roots are: . and . . . . in terms of
Since are roots of: .
. . then: .
The sum of the two roots is:
. . .
Substitute  and : .
. . Hence, the -coefficient is: .
The product of the two roots is:
Substitute : .
. . Hence, the constant term is: .
Therefore, the equation is: .