Lenovo ThinkPad X250 internal battery replacement

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Turn off your device completely and disconnect it from the power supply.
If there is an SD card in the SD card slot, remove it to make it easier to open the device.
Remove the externally accessible battery.

Take a Phillips PH1 screwdriver and use it to loosen the 8 screws that secure the bottom of the case.
These are captive screws, meaning they can only be loosened but not removed.
The bottom of the case is additionally secured with plastic clips, especially on the side where the hinge is located (top of the picture, marked in orange) and on the side where the headphone jack and SD card slot are located (left in the picture).
Carefully pry the case apart on the other sides where there are no clips. Then run a pick along the gap to release the clips. This may make a loud cracking sound.
Remove the bottom case and set it aside.

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Magnetic Project Mat

$19.95

Take a Phillips PH1 screwdriver and use it to remove the three screws that secure the battery.
These are not captive screws, so store them safely, such as on a magnetic mat, so you don't lose them.
Remove the battery and place it in front of the case.
It's still connected by a cable, so don't try to remove it completely.

Now disconnect the battery cable connector from its socket. To do this, gently move the black plastic part containing the colored cables back and forth, slowly pulling it out of the socket.
When installing the new battery, make sure you plug in the connector the right way round. On my battery, the red cable was on the far left, so the new battery's connector also had to be plugged in with the red cable on the left.