Hint : evaluate the determinant of the matrix whose columns are those three vectors. Solve for the values of for which this determinant is nonzero. (Do you see why?)
How can you possibly get in there? You should get a second degree polynomial in .
This works, because for the three vectors to be a basis of they should form the sides of a parallelepiped having nonzero volume, and the determinant is that volume. Or, more abstractly put, if is a basis, then the change of basis transformation from to the standard basis is invertible; its matrix representation is the one whose determinant you evaluated (with a small mistake).