connecting a hp laptop charger adapter three wire output to a two wire cable
If you do not use a multimeter to verify that you made the right connections regarding + and -, you risk damaging the laptop.

Please consider if that is an acceptable risk to you. It is much less risk to simply get a new adapter. New adapters are almost always cheaper than having your laptop repaired.

The 3rd wire might be used for adapter identification. If your laptop needs that identification then it will not charge the battery when you're using the 2-wire adapter.

Again a reason to get the proper charger and not fiddle around with this.