Low beam and running lights will not come on

I DON'T KNOW. $1000 ASSEMBLIES ARE PROFIT MAKERS FOR DEALERS. I THINK ASSEMBLIES ARE JUST PLASTIC LIGHT SHADES AND NOT RELATED.
VISUALLY COMPARE PART NUMBER OR LOOK OF OLD BULBS WITH NEW BULBS. CHECK WITH CAR MANUAL.
WHAT IF THE LIGHT SWITCH OR RELAYS ARE FAULTY.?
AT NIGHT LOOK AT RUNNING LIGHTS. HAVE SOMEONE TURN ON LOW BEAMS.THEN HIGH BEAMS TO BE SURE.
YOU HAVE 3 LIGHT LEVELS. YOUR RUNNING LIGHTS AND HIGH BEAMS ARE WORKING BUT NOT YOUR LOW BEAMS. LOOK IN YOUR OWNERS MANUAL FOR THE HEAD LIGHT FUSES AND OR RELAYS TO SEE IF THERE ARE SEPARATE ONES FOR HIGH AND LOW BEAMS.

Do you know anything about automotive electronics ? Electronic control modules .
Headlamp ON/OFF control is determined by the body control module (BCM) by a signal on the headlamp switch on input circuit when the headlamp switch is in the HEAD position. When the headlamp switch is in the AUTO position, the BCM determines headlamps ON/OFF by the voltage from the ambient light sensor.
If the headlight switch is left in the ON position, the inadvertent power control feature will turn OFF the headlights approximately 10 minutes after the ignition switch is turned to the OFF position. If the driver places the headlight switch in the ON position after the ignition switch has been turned OFF, or if the ignition switch is in the ACCY position, the headlights will remain ON until turned OFF or the battery runs dead.
The high beam indicator is illuminated when the instrument cluster receives a GMLAN serial data message from the BCM that the high beams are illuminated.
High Beams
The high beam headlamps receive voltage from the headlamp high relay located in the underhood fuse block. The BCM controls the relay on the headlamp high beam relay control circuit. High beam control is determined by a signal on the headlamp dimmer switch signal circuit from the multifunction switch. When the headlamp dimmer switch signal circuit is grounded, the BCM grounds the headlamp high beam relay control circuit, illuminating the high beam headlamps. The right and left high beam circuits are independent and are fused independently in the underhood fuse block.
Low Beams - Standard
The low beam headlamps receive voltage from the headlamp low beam relay located in the underhood fuse block. The BCM controls the relay on the headlamp low beam relay control circuit. Low beam control is determined by a signal on the headlamp dimmer switch signal circuit from the multifunction switch. When the headlamp dimmer switch signal circuit is open, the BCM grounds the headlamp low beam relay control circuit, illuminating the low beam headlamps. The right and left low beam circuits are independent are fused independently in the underhood fuse block.

Hooking up a factory scan tool an checking DTC'S - diagnostic trouble codes would be the very first step in trouble shooting this problem .
DTC B257A 00: Headlamp Switch Input Signal Mismatch
DTC 2575 00: Headlamp Control Circuit

No on the fuse . The LH HDLP and RH HDLP fuse in the fuse block - under hood, supply battery positive voltage to both low and high beam lamps at all times. For manual operation, when the headlamp switch is in the ON position, the headlamp switch supplies ground directly to the headlamps and to the body control module (BCM) on the headlamp low or high beam signal circuit, depending on the position of the headlamp dimmer switch. The BCM uses this signal to determine if the conditions are present to illuminate the daytime running lamps or park lamps.
The headlamps may be turned on 2 different ways. First, when the driver places the headlamp switch in the ON position, for manual operation. Second, with the headlamp switch placed in the OFF position activating automatic lamp control (ALC).For ALC operation, the BCM receives a signal from the ambient light sensor indicating a low or high ambient, outside, light level. Under low light conditions the BCM grounds the headlamp relay control circuit energizing the headlamp control relay. The now closed relay supplies ground through the headlamp switch to the headlamps and the BCM on the headlamp low or high beam signal circuit, depending on the position of the headlamp dimmer switch. When the BCM grounds the headlamp relay control circuit it also grounds the park lamp relay control circuit illuminating the park lamps. The headlamp relay is grounded at G203.
Daytime Running Lamps (DRL)
The low beam headlamps operate at reduced intensity when in daytime running lamps (DRL) mode. The ground circuit to the switch side of the DRL relay has a resistor in-line to reduce the intensity of the low beam headlamps for DRL operation. The body control module (BCM) receives a signal from the ambient light sensor indicating a low or high ambient, outside, light level. Under daylight conditions the BCM grounds the DRL relay control circuit energizing the DRL control relay. The now closed relay supplies ground directly to the low beam headlamps and to the BCM on the headlamp low beam signal circuit. The low beam headlamps now illuminate with reduced intensity. The DRL relay is grounded at G203. Any time the headlamps are illuminated the DRL will not illuminate. Manual park lamps do not effect DRL operation.

Replace the turn signal switch. All exterior lights run threw it except the high mount brake light.

on the 2005 it was on the left side of steering column low on the dash panel.

This is usually caused by a failed front control module as this unit controls the lighting on the vehicle!
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airbag reset tool

just replace the dimmer switch under the column its wore out on the low beam side

The Equinox should still be under warranty so take it back to the dealer, but if it's out of warranty. It's most likely the wiper switch on the stock handle which also controls the headlight hi and low beam. I believe you can buy this part at Auto Zone or Part Source, but if it's a warranty thing, do the dealer. Good luck, Auto Zone and Parts Source will have the step by step instructions for you to replace the part.

Might want to check the fuses if you haven't already.

Possibly replace the bulbs themselves. Mine failed too and the dealer traced it to non-working bulbs after $125 spent. (Annual wallet flush?)