Ok so this concept is kind of hard to explain, but basically the two equations are equal because you are adding 9/4 and then subtracting 9/4, so you just end up with 0. The reason you do that is so you can make the (x + 3/2)^2 part, which you wouldn't be able to do without the 9/4/
When changing something to Vertex form you have the, in this case, coefficient to 3x which is 3 then square it. So you half it to 3/2 then square the fraction which becomes 9/4. Because you are adding the 9/4 in order to make this a perfect square you have to subtract it on the outside too leaving you with the answer provided.
In short:
When changing to stand form you need to half and square the X term to get the x intercept.
