Question 1

Will TP-LINK RE650 AC2600 Wi-Fi Range Extender work with Verizon Fios Quantum Gateway Router?

Accepted Answer

This is also an Access Point, as long as another network host(Router) provides DNS , DHCP and routing services, WiFi devices can connect to this device and be able to communicate on the network and to the outside network(Internet).   Of course you have to connect the Ethernet cable from the Router to the RE650 Ethernet port.

I do not recommend using the Range Extender feature if router(if its a WiFi router) provides bad WiFi reception, extending  a bad WiFi reception by a range extender is still bad WiFi.

Question 2

The internet provider said to get a docsis 3.0 extender. Will this work?

Accepted Answer

This is not a docsis extender.  This is Wireless Range extender and Access point.  If your on Range Extender mode, this will extend the current wireless signal further.  If this is on Access point mode, it will need some other device providing essential services like routing, DNS, and DHCP. A router(usually) provides these services.  The best example of Access point mode in use, is in hotels that provided wired Ethernet(from the wall).  Then you plug that wired Ethernet to RE650 and now you can have wireless devices connect to RE650 as if  its a regular router.

Maybe your looking for a combo cable modem/router?  Docsis Modem will only provide one actual wan IP.  So that means only one device can get on the Internet.  Now your asking I want my other devices to get on the Internet including computers and tables that do not have wired Ethernet connections.

This is where the combo cable modems/routers come in.  I won't get into specifics of how it works but know this.  Combo modems/routers come with wireless functions so tablets and other wifi devices can connect to the Internet  Also they come with Ethernet ports on them for wired devices to connect to them(like computers).  Also you connect your cable wire to this device as well.

Though your question is not clear still.

1.  If you have a combo modem/router and want to extend your wireless signal to other parts of the house?

Answer:  Yes? This will extend your current wifi signal and does not have to be connected physically to your combo cable mode/router.

2.  If you have a cable modem and only one device can get on the internet(only a wire connection)

Answer:  Yes? Then you need to get a combo cable modem/router

Unlikely scenario.
3.If you have combo cable modem/router but does not have wifi or it does have wifi but the wifi is not great and want to replace the wifi with a better wifi device.

Answer:  Yes?  You can use this but it will connect to the combo cable modem/router behind the device.

Brand	TP-Link
Model	RE650

Standards	IEEE 802.11ac, IEEE 802.11n, IEEE 802.11g, IEEE 802.11b, IEEE 802.11a
Frequency Band	2.4 GHz / 5.0 GHz
Wireless Data Rates	5 GHz: Up to 1733 Mbps 2.4 GHz: Up to 800 Mpbs
Security	64 / 128-bit WEP WPA-PSK / WPA2-PSK
Antenna	4 x external
Transmit Power	<20 dBm (2.4 GHz) <23 dBm (5 GHz)
Receiving Sensitivity	5 GHz: 11a 6 Mbps: -96 dBm @ 10% PER 11a 54 Mbps: -78 dBm @ 10% PER 11ac HT20 mcs8: -69 dBm @ 10% PER 11ac HT40 mcs9: -66 dBm @ 10% PER 11ac HT80 mcs9: -63 dBm @ 10% PER 2.4 GHz 11g 54M: -79 dBm @ 10% PER 11n HT20 mcs7: -77 dBm @ 10% PER 11n HT40 mcs7: -74 dBm @ 10% PER
Interface	1 x 10/100/1000M Ethernet Port (RJ45)
Button	WPS Button, Reset Button, LED Button, Power Button
Power	12W (max power consumption)
System Requirements	Microsoft Windows 98SE, NT, 2000, XP, Vista or Windows 7, 8, 10, Mac OS, NetWare, UNIX or Linux
Dimensions	1.60" x 6.40" x 3.40"

Operating Humidity	10% ~ 90% non-condensing
Storage Humidity	5% ~ 90% non-condensing

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TP-Link AC2600 WiFi Extender(RE650), Up to 2600Mbps, Dual Band WiFi Range Extender, Gigabit port, Internet Booster, Repeater, Access Point, 4x4 MU-MIMO

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Item#: 9SIC1B3KKC4384