begründen
to justify
To present and explain the reasons or causes for a statement, opinion, or action.
Sie müssen Ihre Entscheidung für die Kündigung schriftlich begründen.
You must justify your decision to terminate the contract in writing.

to justify / establish
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to justify
To present and explain the reasons or causes for a statement, opinion, or action.
Sie müssen Ihre Entscheidung für die Kündigung schriftlich begründen.
You must justify your decision to terminate the contract in writing.

The verb 'begründen' has the prefix 'be-' and is inseparable. So the prefix remains firmly attached to the verb in all sentence forms and is not placed at the end of the sentence.
Er begründet seine Entscheidung ausführlich.
He justifies his decision in detail.
The verb 'begründen' is closely related to the noun 'der Grund' (the reason; the ground). This helps in understanding both meanings: you can give reasons for something (meaning 1), or lay the foundation for something (meaning 2).
Können Sie mir den Grund für diese Änderung nennen?
Can you tell me the reason for this change?
When the reason is stated directly, the preposition 'mit' (with) and the dative are often used. Alternatively, you can use a subordinate clause with 'dass' (that) or 'indem' (by).
Der Minister begründete die neuen Regeln mit der aktuellen Sicherheitslage.
The minister justified the new rules with the current security situation.
In the meaning 'to establish', 'begründen' is often used with abstract concepts that have a beginning. Typical examples are: 'eine Freundschaft begründen' (to establish a friendship), 'eine Tradition begründen' (to establish a tradition), or 'einen Ruf begründen' (to establish a reputation).
Ihre Zusammenarbeit begründete eine lange und erfolgreiche Partnerschaft.
Their collaboration established a long and successful partnership.
to explain / declare
'erklären' and 'begründen' can both have to do with reasons and understanding. 'erklären' is more general: you make something understandable, for example a task, a rule, or a process. You use 'begründen' when you explicitly give reasons for an opinion, decision, or action. Learners may confuse the words because a good justification often also explains something. The difference is: 'erklären' often answers 'How does this work?' or 'Why is this the case?', while 'begründen' is more likely to answer 'Why did you say or do that?'.

Beim Termin hat mein Kollege seine Entscheidung mit klaren Zahlen begründet.
At the meeting, my colleague justified his decision with clear figures.
Sofia erklärt ihrem Bruder die Aufgabe noch einmal langsam.
Sofia explains the task to her brother again slowly.