Aaaaah Sony Vaio!!?
Answers:
I suggest you take it to a laptop repair shop for an accurate diagnosis and cost estimate, just so you know what is really wrong with it and you can better choose what to do. There are multiple options as to what can be wrong, from a fried CPU to a loose wire.
My Vaio laptop had repeated problems with the fan but it didn't stop the screen from displaying. And I don't recommend using Sony's repair service - I didn't get mine back from a warranty repair for 4 months (just long enough for the warranty to expire so they could refuse to repair it again even though they didn't fix it the first time).
I'm using a Toshiba laptop right now... the Vaio laptop is sitting by our broken Vaio desktop computer, waiting to be torn apart for parts. Our new Gateway desktop will be arriving in two days. I'm sure you noticed that we didn't buy Vaio's again ;-)
I can tell you right now that I would love to help you with this, but I don't think you are going to get the correct diagnosis from anyone here.
There isn't much that can be done to help you diagnose the problem when the screen is not functional.
You should bring your laptop into a small repair shop and ask if someone can diagnose the problem and give you a price quote to fix it. At least that way you can know what the problem is and then ask the community for help to fix it. If the problem was something that was a design flaw that Sony decided to pay for all customer repairs then well.Bonus!
The $700 price quote you got from Sony is probably their robot response to all unknown laptop problems.
I do not know where you are but if you have a Fry's electronics's close take it in to service department, they will look at and advise for free on needed repairs. The only time I have had a similar problem my hard drive crashed, I went to Fry's they check, sold me a new hard drive, showed me how to install and reinstalled recovery CD and now has worked like new for about two years.
Hope this helps
Sony probably wants to replace the whole LCD which would explain why $700.
Take it to a local repair center, they hould be able to test backlight, and inverter, or maybe for loose cables.
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