XPS 15 L502X would not boot, display pink, 8 beeps

@JOcean pointed you in the right direction, likely a display or display cable issue (or maybe the display grounding cable, which you should also check).

Laptops receive a lot of abuse that desktops don't. Even when treated gently, thousands of opening and closings of the display/lid can cause a display cable to pinch or its connector to loosen. That's especially true if the laptop was ever opened for internal work and the display cable not properly routed or solidly connected.

Examine and reconnect display cable and ground cable

I would open the laptop to examine the display cable. Its connector lifts straight up vertically by holding and pulling on the flexible plastic tab/loop from above. This Parts-People video shows disconnection of the display cable and display removal. But, treat the cable more gently than the guy in the video did. Also, check the display grounding cable, as I mentioned.

Cleaning and cooling

As that laptop is 12 years old, the thermal compound has dried out and cannot adequately transfer heat away from the processor, and there may be a lot of dust inside the laptop and maybe inside the fan and heat fins. Your laptop is unable to adequately cool itself.

You might consider an internal cleaning. Open it up to clean out dust. Disassemble the fan to clean out dust and fluff. I found a fair bit in two laptops I worked on recently. Remove the heat sink, remove the old thermal compound and apply new thermal compound before reinstalling the heat sink. On my XPS L502x, I disassembled the fan in order to thoroughly clean it. It now looks like new.

While you're in there, replace the CMOS battery. If it isn't dead, it soon will be. BIOS settings and date/time can't be saved without a functioning CMOS battery.

Follow the disassembly and reassembly steps in Dell's Service Manual.
Look on YouTube for videos showing disassembly of your model. But, be cautious with them, as some use steps or methods that are not appropriate. Use the videos to add to your understanding of what to do, but follow the steps in Dell's manual.

Whenever touching components or working inside a computer, wear a grounded wrist strap, also called anti-static wrist strap, ESD wrist strap, or ground bracelet. I know many people do such work without this, but it's a cheap and sensible precaution. Rest the laptop on an anti-static mat or at least a reasonable alternative such as corrugated cardboard.

If you find there's a catastrophic failure of the display, don't dispose of the laptop. I'll gladly arrange to get some parts and screws from you as backups for mine

Stop using Windows 7

I have the identical laptop, XPS L502x. I've worked on the interior components and streamlined Windows, and it runs well. I cleaned everything, applied new thermal compound and thermal pads, installed an SSD, clean-installed Windows 10, etc.

Windows 7 is unsupported and insecure. It does not receive security patches, so it should no longer be used on a PC connected to the Internet.

I suggest that you update to Windows 10. Do the free upgrade: https://www.zdnet.com/article/heres-how-you-can-still-get-a-free-windows-10-upgrade/

After upgrading, check to see if Windows 10 is activated in Settings | Activation. If it is, then you can perform a clean install of Windows 10 and it should activate automatically.