Why is the power adapter will not power up my laptop

Did you ever change the battery? These are from 2010 just based on the 1st generation Core i processor - unless the battery was replaced recently, they’re pretty much all dead outside of the well maintained packs and even those are done after 5-6 years as everyday batteries. BUT before you go out and buy a new battery, try booting without one installed and see if it powers on. This usually solves the issue because it’s often a bad cell in the pack - usually the 1 or 2 that didn’t age evenly with the rest of the battery. Yes, one bad cell that didn’t consistently age with the rest of them in a 6 cell pack can kill the whole pack as a safety measure.

If it doesn’t boot without the battery, get a multimeter (even a cheap one from Walmart will work - but don’t use it on AC!) and test the power adapter to make sure it’s still good. On these it’s 19.5-20V nominal/19.5V load. The inside jacket is positive and the outside jacket is negative, so it uses a center positive adapter.

IMPORTANT: IF you find the adapter has issues, please don’t buy a knockoff from eBay as tempting as it is for a 10 year old laptop - at least purchase a used OEM adapter. It’s still a gamble, but it’s far less of a gamble as long as you stick to buying the modern adapter Dell sells at retail. If you are looking at the retail adapter without a box sold as a new adapter (or it’s sold in “bulk packaging”), it’s usually fake or a misleading used adapter; DO NOT BUY THESE. If you’re trying to keep costs down and want to save a few bucks, there are 2 other styles that are easy to find, but one is best avoided.

• AVOID: These early adapters (PA) are from ~2004-2008 when you could still buy a new Core Duo/Core 2/Pentium laptop over 10 years ago. These are more prone to capacitor wear, which will reduce the tip voltage (the computers often do not like it and have problems with the adapters). These may not change the battery as well since they are so old! The tip voltage issue is often not an issue with minor voltage loss (0.5V) on lower end platforms but it tends to be a problem as it hits 1-2V or it's used on a higher end system. I have seen them last but those even tend to only last so long. I wouldn’t rely on it day to day (or buy one, but if it came with the laptop I’d take it and test it) but if I had a good PA adapter I’d use it in a pinch.
• BUY (But spend as little as possible): For the native Core i series machines, they switched to a different style OEM adapter (DA). These are the better used budget option, but are a dime a dozen. Do not pay more then $15 with shipping at the most. Most eBay sellers will sell you a few for similar money as a lot since they’re so common. They typically come from No HD laptops where it was shipped with a bunch of adapters and they bought a lot, but sold the computer and adapter separately.
• BUY: Dell curved charger (FA/HA/Others). Most expensive, but best built adapter. It is less prone to insulation failure then the OEM and PA adapters, but tends to cost more even used.

If you still have issues booting with a new power adapter and battery, it’s usually a motherboard issue. Honestly, your money is better spent on a retired business 8th-10th gen machine (with an SSD) over a new board for this laptop if the cheap fixes make no difference if you cannot afford a new one, even if it’s the “common” i5/8GB/Intel video version. Before scrapping the laptop at this point, try a new CMOS battery.