I got rid of the cockroach from the room = 部屋からゴキブリを追い出した
cockroachはゴキブリを意味します。また、get rid (got rid)が動詞であり、処理する・払い出す・退治するという意味となります。get ridが現在形、got ridが過去形です。
例）He was bothering me, so I got rid of him＝彼が迷惑だったから払い出した
To chase something means to go after or to follow so you would push along the cockroach out the room
By getting rid of or removing something you are making it not be in that place anymore so for example you removed the cockroach from the room by picking it up with tissue
There are a number of ways to describe the act of removing cockroaches from your room. You can use formal language such as
I removed it (the cockroach) from my room.
You can be a little less formal and say 'I had to get rid of it.' rid meaning to remove it, or you could try a little informal humour and describe the event as a fight between you and the cockroach which resulted in you winning.
部屋からゴキブリを追い出したことを表す場合、いろいろな言い方ができます。
以下のようにフォーマルに伝えることもできます。
I removed it (the cockroach) from my room.
部屋から（ゴキブリを）追い出しました。
もう少しインフォーマルに、以下のようにも言えます。
'I had to get rid of it.'
ゴキブリを追い払わないといけなかった。
rid とは取り除くという意味です。
また、ゴキブリと戦って勝った、というようにインフォーマルにユーモアを交えて伝えることもできます（三つ目の例）。
The two sentences you see above are great ways to describe that you chased a cockroach out of your room. In the first sentence you will see the verb run and in the second sentence you will see the term scare off. Although the word run and the term scare off have different meanings, in these sentences they mean to cause something or someone to go in a particular direction.
A casual way of describing the situation would be to say, ‘I got rid of the cockroach’ which means you got it away from the room. However, this doesn’t describe the method by which you did this, e.g. you might have killed it or thrown it out of the window. Therefore, a better way to make the statement is to say ‘I chased it out of the room’, showing that you coaxed it to leave the room through the door.
この状況をカジュアルに表すなら、「I got rid of the cockroach.」と言えます。ゴキブリを部屋から追い出したという意味です。ただ、これは、どのように追い出したかについては言っていません。例えば、殺したのかもしれませんし、窓から投げたのかもしれません。ですから、より良い言い方は「I chased it out of the room.」です。これは、部屋から出て行くように誘導したことを表します。
カナダ 
アメリカ合衆国 
イギリス 
