IPhone 4 only turns on when plugged in?

I have an iPhone 4 and it only turns on when plugged in. I dropped it one day and it wouldn't turn back on. I know for sure it's not stuck in DFU mode. When I plug it into my wall charger it will turn on and it will take me to an iOS 6 setup saying slide to setup. I do so and it just says connect to iTunes. I attempt to connect to iTunes and it just flashes on and off. The Apple logo will come and than it will turn off. I called FastFix in the mall and they told me it was probably the battery, But, I'm not paying $50 dollars to have it checked out. That's just the biggest rip off. I'll only pay for a repair. I need an answer. I miss my phone and I want it back. Please only answer if you know what you're talking about. It's not the charger, it's not the port. I just want to know for sure if it's the battery. Thanks.

Added (1). It dies as soon as I unplug it.

If it only runs when plugged in, and dies immediately upon severance of power, the battery (or it's connection to the phone motherboard) is most certainly the fault.

If you feel comfortable with doing minor-scale "surgery" on your phone, there are battery replacement kits out there for your phone that are typically less than $30 that include the battery, the needed screwdriver, and other tools.

There are videos on YouTube that show how it is done. So much as you have the right tools on hand, and feel comfortable with the replacement, it can be done in about 5 minutes. I have included a link to a video in the source section below.

Just be aware that if you or a non-Apple store replaces the battery, your warranty will be voided out (which is likely the case here anyway, due to the phone's likely age).

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