regeln
to handle
To solve a problem, organize a matter, or take care of a task.
Keine Sorge wegen der Tickets, das regele ich.
Don't worry about the tickets; I'll handle it.

to handle / regulate
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to handle
To solve a problem, organize a matter, or take care of a task.
Keine Sorge wegen der Tickets, das regele ich.
Don't worry about the tickets; I'll handle it.

The verb 'regeln' is derived directly from the noun 'die Regel' (the rule). So it literally means to bring something into a state of order or in line with a rule.
Eine Ampel regelt den Verkehr an der Kreuzung.
A traffic light regulates the traffic at the intersection.
The reflexive form 'sich regeln' means that a problem or situation solves itself without anyone having to intervene. It is often used together with 'von selbst' (by itself/on its own).
Mach dir keine Sorgen, das wird sich schon von selbst regeln.
Don't worry, it will sort itself out on its own.
In the meaning 'organisieren' (to organize) or 'erledigen' (to take care of), 'regeln' is very common in spoken language. The sentence 'Ich regel das.' is a common way to say that you will take care of something.
Wer holt die Getränke für die Party? – Keine Sorge, ich regel das.
Who's getting the drinks for the party? — Don't worry, I'll handle it.
When technical devices are involved, 'regeln' always means 'einstellen' (to set) or 'anpassen' (to adjust). Typical objects are 'Temperatur' (temperature), 'Lautstärke' (volume), or 'Geschwindigkeit' (speed).
Kannst du bitte die Heizung etwas höher regeln? Mir ist kalt.
Could you please turn up the heating a little? I'm cold.
In formal or legal texts, 'regeln' means that something is bindingly established by a law or regulation. The subject here is often the law itself.
Die Straßenverkehrsordnung regelt die Vorfahrt an dieser Kreuzung.
The traffic regulations govern right of way at this intersection.
to solve / loosen
'lösen' and 'regeln' can both mean that a problem no longer exists. 'lösen' emphasizes more that you find the right answer or a solution, for example with an exercise, a puzzle, or a concrete problem. 'regeln' emphasizes more that you take care of a matter and organize or deal with it. For a math problem, 'lösen' sounds more natural. For an appointment, an argument, or a practical matter, people often say 'regeln'.

Den neuen Termin regeln wir morgen im Büro.
We'll handle the new appointment at the office tomorrow.
Lisa löst die schwierige Aufgabe, weil sie lange darüber nachdenkt.
Lisa solves the difficult problem because she thinks about it for a long time.