Actually no new computer is needed. Believe it or not Pentium 4's are still solid workhorses and can keep up with most of the stuff we run today. After we hit 2Ghz processors in 2000, things started to slow down from the whole 90's rush of technology.

I own a Pentium 4 2.4Ghz and runs Minecraft just fine, I can even host a server off it with 5 people. You're actually lucky to have a 3.8Ghz Pentium 4, they are one of the highest speeds they had and could be over-clocked to 4Ghz and are somewhat rare in garage sales. Pentium 4's (and perhaps Pentium D's to some extent) are also sort of a novelty by Intel fans, as that generation never continued and future generations were based on Pentium 3 mobile processors to keep it cooler. DirectX will give you some issues depending on the integrated graphics, but will run smoothly when you use an OpenGL based OS, one based on Linux in other words. I have one Pentium D that has given me problems with Linux based OS's (even Windows XP) with severe lag playing Minecraft. Pentium D was a really bad experiment gone wrong.

Yeah, you're really going to need a new PC. Fair enough, a 3.8GHz processor is fine, but it's an Intel Pentium 2. In addition, you're running an iGPU, so you don't have a dedicated graphics card to play games. You said that it's a bad card- what is? Do you have a dedicated GPU which just isn't working? If so, just contact the manufacturer/wherever you bought it from about getting it RMA'd. You're going to need a new OS, too. In fact, the only thing that probably worth keeping is the case. You'd be better off buying a relatively cheap gaming PC. You could pick up at least an FX-6350 (OC it to 4.2GHz minimum), an R7 260X, 8GB RAM, an MSI MOBO, a 500W PSU, and a decent case for around £500.

Your graphics card is integrated into the CPU (Intel HD Graphics). With that old of a card and such a small amount of VRAM, don't expect to run Minecraft smoothly whatsoever. Newer IGPUS like HD Graphics 4600 can run Minecraft at good framerates, but you should really get a dedicated GPU, like a GTX 780 or R9 if you want really good performance on all games.

You need a new PC. Badly.

How do I put an image on here?

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Ill ask my local IT for one

I found out the problem: Bad video card

Purchase a dedicated graphics card that does not have external power requirements, then save up some money for a new PC and come back here and ask about what parts to get, and we will help you out!

I purchased a Radeon HD 7750 from XFX, very good card, plays MC on max settings at 60fps, you can pick one up for about $100.

I advice getting a new computer, I'm surprised your's can even handle it.

My secondary computer is a GX620. I have no problems running it.

DO NOT USE ITS INTEGRATED GRAPHICS, it can barely run windows 7 smoothly, I recommend adding a video card to it so you dont have integrated graphics. I put in a AMD Radeon HD 6450 2GB card. Did it improve performance, yes it did, I was able to run minecraft at low setting at 60-100 FPS, but until recently, around 30-60 with 1.8. What I recomend is you install Windows 7 or 8, cant afford it, pirate it, its easy and you will get better performance, this is what I did when I got it two years ago for free, I got it to play some games. Remember I dont use it as my primary, i use it when I have a lot of stuff going on my main desktop.

EDIT: You wont get much performance in gaming with an older Optiplex, Optiplexes are built for business' where people dont play games like minecraft

EDIT: Plus they are so old, i think the 620s were realeased 9 years ago

You must have an iGPU. My school actually uses these, and they can _BARELY_ run flash games. Same processor and gpu. You're computer probably can not handle minecraft, unfortunately.