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The computer is plugged into a working outlet, but does not charge.
Unfortunately, there is no indicator on the charger to see if the charger is working. If you or a friend has another compatible laptop, test to see if the charger is working. If the charger is broken, it will need to be replaced.
Try removing the battery and try to run the laptop while plugged in. If this works, then the battery needs to be replaced.
The power adapter inside the laptop may have broken, or it may be some other internal failure. (Some possible underlying causes are faulty motherboard, bad logic board, damaged charging circuit, malfunctioning battery sensor, etc.)
The computer is running unusually hot, and performance may be affected.
A buildup of dust and dirt will block air flow and cause overheating. To fix this, take a can of compressed air and clean out the inside of the case. Be sure to clean all dust from the fans and the heat sinks on components.
Thermal paste helps maintain a good contact between the CPU and it's heat sink. If the heat sink has been dislodged or bumped (possibly while replacing a component) it could cause your CPU to overheat. This video shows how to apply fresh thermal paste.
If you notice that your fans aren't spinning while the computer is at high temperature, they may need to be cleaned or replaced.
There is a blemish on the screen that won't go away.
LCD crystals can become stuck or dead, resulting in them failing to change color with the rest of the screen. Dead pixels will appear to be either solid white or solid black, while stuck pixels display light but only red, green, or blue on your screen. There are websites like this one to help you determine if you have stuck or dead pixels.
LCD screens use liquid crystals to display images. However, liquids can sometimes get solidified, causing them to be stuck and not change color. Pressing gently on the screen can unstick a pixel, but it can also create more. Start by tapping on the dead/stuck pixel, and if that doesn't work gently apply pressure.
There are a few pieces of software out there that will attempt to fix dead/stuck pixels by rapidly cycling them through different colors. By turning pixels on and off using a software, dead/stuck pixels can potentially be re-energized. LCD Repair is one of these softwares and is web based. It should work regardless of platform, but programs for different operating systems also exist.
If the methods listed above don't work, you may need to replace your screen.