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I'm thinking if there is no video on the laptop screen and the external monitor then the video card has failed possibly due the fan fan failure. <br />
Some Elite books had removable video cards, not sure about the 850.
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If the video card has failed it does not look good for replacement, see this link from HP:<br />
<a href="https://h30434.www3.hp.com/t5/Notebook-Hardware-and-Upgrade-Questions/GPU-upgrade-options-for-HP-EliteBook-850-G3/td-p/6563338" rel="ugc">https://h30434.www3.hp.com/t5/Notebook-Hardware-and-Upgrade-Questions/GPU-upgrade-options-for-HP-EliteBook-850-G3/td-p/6563338</a>
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Turn the computer on and immediately press F1 repeatedly. &nbsp;Can you get into BIOS? &nbsp;If nothing, the issue may be more than just the fan. <br />
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You do not want to replace random parts, so what about taking it to a local repairman for an estimate and then determine if it worth repairing
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Highly unlikely that a fan problem would cause those symptoms as soon as the computer is powered up.<br />
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Does the computer have LEDs for CapsLock or NumLock? &nbsp;If so, do they change when those buttons are pressed? &nbsp;LEDs changing will tell you that the system is processing. &nbsp;If they don't, then the system isn't really running.<br />
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Another test is to connect an external USB keyboard and watch its LEDs when the system powers up. &nbsp;They should flash on and then off. &nbsp;If they do not, the system isn't getting through the initalization process, indicating a likely motherboard problem.
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here my laptop troubleshooting guide, i hope it helps you :<br />
- remove all external devices<br />
-you can disable devices from BIOS to test (sound, lan, etc..)<br />
- if no display - test with an external monitor<br />
- test if it runs from battery alone, or from AC alone<br />
- test also with a known good battery !<br />
***some laptops have a reset button - if you have one use it!<br />
- remove battery and AC adaptor, hold power button for 20 sec, connect AC and reboot --&gt; any display ? if NO --&gt;<br />
- disconnect all disk and cd drives, battery , and other cards and peripherals, leave only 1 ram stick<br />
- boot --&gt;any display ?<br />
-if you can access the bios battery, remove the battery, AC, and bios battery for a minute to reset the bios<br />
or test it, it should read 3 V<br />
*** you can test the disk on another pc, if you like !<br />
<br />
===&gt;&gt;you can test if the mobo works outside the casing, connected to power and external video &nbsp;!<br />
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you can also remove and reseat the video plug - it sometimes comes loose
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CompProbSolv....<br />
<br />
Can't get to the BIO by pressing F1, but LEDs do register when pressed on internal and external keyboard....what does that mean.
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Thanks for always being to the point.
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If the LEDs switch state when you hit the NumLock or CapsLock keys, that indicates that the CPU is functioning. &nbsp;The keyboard doesn't keep track of the proper state of the LEDs. &nbsp;When you hit the key, that gets processed by the CPU which checks the current state (it's held in memory), flips it, and then sends the appropriate command to the keyboard to turn the LED on or off.
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@nplanek &nbsp;- Thanks. &nbsp;There may be another hardware issue here (that is, not just the CPU)
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I have a HP Elitebook 850 G3 running Windows 10 that won't display anything (hooked to an external monitor no display also). &nbsp;It has power because the keys light up. &nbsp;I've tried switching memory cards with no luck. &nbsp;The fan is not working. &nbsp;Would a fan not working cause the no display issue? &nbsp;I don't what to purchase a new fan if the problem is something else.
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I have a HP Elitebook 850 G3 running Windows 10 that won't display anything (hooked to an external monitor no display also).  It has power because the keys light up.  I've tried switching memory cards with no luck.  The fan is not working.  Would a fan not working cause the no display issue?  I don't what to purchase a new fan if the problem is something else.

No Display - Dead Fan

Windows 10 is a personal computer operating system featuring the "universal application architecture" (UAP); apps can be designed to run across multiple devices with nearly identical code, including PCs, tablets, smartphones, embedded systems, Xbox One, Surface Hub and HoloLens. Windows 10 also includes a virtual desktop system, a window and desktop management feature called Task View, the Microsoft Edge web browser, support for fingerprint and face recognition login, voice-based search (Cortana), new security features for enterprise environments, and DirectX 12 and WDDM 2.0 to improve the operating system's graphics capabilities for games.

Windows 10

This topic area includes legacy versions of Windows prior to Windows 2000: Windows 3/3.1, Windows 95 and Windows 98, plus any other Windows-related versions including Windows Mobile.

Windows OS

A laptop or notebook is a portable personal computer with a clamshell form factor, suitable for mobile use. Although originally there was a distinction between laptops and notebooks, the former being bigger and heavier than the latter, there is often no longer any difference. Laptops are commonly used in a variety of settings, such as at work, in education, and for personal multimedia. A laptop combines the components, inputs, outputs and capabilities of a desktop computer, including the display screen, speakers, a keyboard, and pointing devices (such as a touchpad or trackpad) into a single unit. The device can be powered either from a rechargeable battery or by mains electricity from an AC adapter.