Mr.maniac said:
could someon give me general instructions about these kind of questions?
The general method is just algebraic elimination, of which you have certainly already met other examples in other simultaneous equations.
It is not necessary, but at this stage may be helpful in overcoming the mental blockage* that is preventing you seeing how easy and natural it is to proceed, if you call the particular value of x that makes a couple of equations both true another name - you could call it x
1 or you could call it α.
Thus you are being told that there exists a number, α for which both
2α
2 +kα-5=0 and α
2-3α-4=0
You can surely from these two get a new equation in which α
2 is eliminated and is still true. After which you should be able to go on eliminating until you get an equation in which α does not appear at all.
(There does exist a general expression in their coefficients, called the 'Eliminant' of two polynomials which vanishes when they have a common root or factor, which is 4x4 determinant for two quadratics. But this is just a convenient formulation of the algebraic eliminations you would do anyway. To show things, even things you know, are connected up, the discriminant of a quadratic is just the eliminant of the quadratic and it's derivative.)
(* there is a possible mental blockage of continuing to think of x as 'a Variable', something that could be just anything, Whereas the problem itself has restricted it to something definite even if not yet known. When you're more used to it you can kick away that prop.

)