Depends.  Short Cards will go in but they may need an ugly external power supply.  The 240W power supply in the 3650 seems ok but the 200W power supply in the 3668 doesn't seem sufficient.

http://www.dell.com/support/home/us/en/04/product-support/product/inspiron-3668-desktop/manuals

It is possible to hack a 3650 working by removing the power supply entirely and using adapter and placing power supply OUTSIDE the case.  There is no hack for 3668 6 PIN 200W power supply.

EVGA 700B1   Any power supply REQUIRES the MODDIY adapter. The EPS12v CPU connection is still 4 pin but the 24 pin needs to be converted to 8 pin.

http://www.newegg.com/Product/Product.aspx?Item=N82E16817438066

Dual slot SHORT cards(GTX960/970) See Recommended card below are fine. 12 inch long cards do not fit.  EVGA 500W unit is recommended for a 960.  EVGA NEX 700B1 is recommended for a GTX970.

A MODDIY adapter is needed to convert 24 pin to 8 pin.  The 3668 has 6 pin so there may not be an option to fix this.  I would get a 3650 rather than 3668 because of this alone.

http://www.moddiy.com/products/Dell-Inspiron-3650-PSU-Main-Power-24%252dPin-to-8%252dPin-Adapter-Cable-%2830cm%29.html

Depends.  Short Cards will go in but they may need an ugly external power supply.  The 240W power supply in the 3650 seems ok but the 200W power supply in the 3668 doesn't seem sufficient. http://www.dell.com/support/home/us/en/04/product-support/product/inspiron-3668-desktop/manuals     It is possible to hack a 3650 working by removing the power supply entirely and using adapter and placing power supply OUTSIDE the case.  There is no hack for 3668 6 PIN 200W power supply. EVGA 700B1   Any power supply REQUIRES the MODDIY adapter. The EPS12v CPU connection is still 4 pin but the 24 pin needs to be converted to 8 pin. http://www.newegg.com/Product/Product.aspx?Item=N82E16817438066     Dual slot SHORT cards(GTX960/970) See Recommended card below are fine. 12 inch long cards do not fit.  EVGA 500W unit is recommended for a 960.  EVGA NEX 700B1 is recommended for a GTX970. A MODDIY adapter is needed to convert 24 pin to 8 pin.  The 3668 has 6 pin so there may not be an option to fix this.  I would get a 3650 rather than 3668 because of this alone. http://www.moddiy.com/products/Dell-Inspiron-3650-PSU-Main-Power-24%252dPin-to-8%252dPin-Adapter-Cable-%2830cm%29.html

Thank you for all this information! But I have one question, my power supply said that the max DC output was 240W, and you said my PSU was 200W. Is my PSU actually a 240W, or is that just the mac watt (as stated). I'm aware that this may be a very stupid question

They reduced the size of the PSU on the 3668 to 200W and went from 8 pin to 6 pin connector.  Short versions of the Card will fit but that does not overcome the power supply limitation.   If your 3668 has a 240W power supply then a NON OVERCLOCKED 75W version of the ZOTAC  1050TI or Gigabyte Windforce Radeon RX460 GV-RX460WF2OC-2GD will work fine. The 3650 is a different machine https://www.walmart.com/ip/49455390     https://www.amazon.com/Gigabyte-Radeon-Windforce-Graphics-GV-RX460WF2OC-2GD/dp/B01K1JV83C/

Thank you very much for the help! Now I just have one final question, the ZOTAC GeForce 1050Ti Low Profile is dual slot, and I'm not 100% sure what this means (except that it takes up 2 slots). I have 2 PCI Express slots (only a available) and one PCI Express 3.0 slot. Would a dual slot graphics card work with this? 

PCI data from Speccy:

Slot PCI-E
Slot Type - PCI-E
Slot Usage - In Use
Data lanes - x1
Slot Designation - SLOT1
Characteristics - 3.3V, PME
Slot Number - 0

Slot PCI-E Gen 3
Slot Type - PCI-E Gen 3
Slot Usage - Available
Data lanes - x16
Slot Designation - SLOT2
Characteristics - 3.3V, PME
Slot Number - 1

Slot PCI-E
Slot Type - PCI-E
Slot Usage - Available
Data lanes - x1
Slot Designation - SLOT3
Characteristics - 3.3V, PME
Slot Number - 2

Thanks in advance!

-Andreas

Never mind about the dual slot thing, I forgot to Google first

Now I'm about to buy the ZOTAC GeForce GTX 1050 Ti Low Profile (www.newegg.com/.../Product.aspx and before I mess up I just want to make sure that it is compatible with my hardware. Thanks!

I understand, but I believe that my PSU is 240W it says "Maximum continuous total DC output power shall not exceed 240W" on my PSU. Does this mean that it is 240W? It also says 100-240V~ on it.

I have the same specs as you WastefulNick and my 3668 came with a 240 watt power supply. I know Dell sells a i7 3668 that comes with a GTX 1050 2GB. A rep at Dell told me all Inspiron 3668 desktops come with a 240 watt power supply.

I wanted to install a GTX 1050 Ti but a local computer store tech told me the 240 watts might not be enough for this graphics card. I decided to go with a GT 1030, which has 30 watts of power consumption compared to the 75W of the 1050 Ti. I'm planning on installing one this coming week and will keep you posted.

1050 TI as well as RX460  75w cards work in a 3650 with 240W power supply.  This video shows 3650 with Gigabyte Windforce RX460 

GV-RX460WF2OC-2GD

https://www.amazon.com/Gigabyte-Radeon-Windforce-Graphics-GV-RX460WF2OC-2GD/dp/B01K1JV83C

possible solution:  the rx 460 has a new little brother, the rx550 that sips a mere 50w

According to the Dell Regulatory and Environmental Datasheet (downloads.dell.com/.../dell inspiron 3668,d19m,d19m003,dell regulatory and environmental datasheet.pdf) on page 5, the 3668 has 5 different power supply models all rated at 240 watts. My 3668 desktop came with model H240ES-02 (plugloadsolutions.com/.../DELL_H240ES-02_240W_ECOS 4616_Report.pdf)

This is a good efficient power supply and I have changed my mind about installing a GT 1030. The one I have is the EVGA GTX 1050 Ti (single fan) non overclock card. I will install it tomorrow and report back.

I also wanted to upgrade the GPU on my new 3668 as I’d bought this PC for my son for Xmas and expected him to play games from time to time. I checked with Dell and received the following:

You mentioned that you want to know the power output of your system's PSU

As per our records, your system Model INSP 3668 Desktop would output the following:

•APFC 240W power supply unit (220V/110V)

•NPFC 240W power supply unit (220V/110V)

As a result, I bought and fitted a Zotac GeForce GTX 1050 2 GB Mini Graphics Card and, so far, all seems well.

I realise that the 75w power requirement of the card could possibly cause problems but, having said that, Dell do supply the 3668 i7 version with a GeForce 1050 as an option. Can someone explain this?