
erklären
to explain
To make a situation understandable to someone or explain how something works.
Der Lehrer erklärt den Schülern die mathematische Formel.
The teacher explains the mathematical formula to the students.
to explain / declare
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to explain
To make a situation understandable to someone or explain how something works.
Der Lehrer erklärt den Schülern die mathematische Formel.
The teacher explains the mathematical formula to the students.

to declare
To announce something officially or solemnly; to make a formal statement.
Die Politikerin erklärte ihren Rücktritt aus allen Ämtern.
The politician declared her resignation from all offices.

to be explained
Used reflexively to express that something has a reason or becomes understandable.
Sein plötzlicher Ärger erklärt sich durch den großen Stress, den er gerade hat.
His sudden anger is explained by the great stress he is currently under.

to declare ...
To officially assign a certain status or quality to someone or something.
Nach langer Diskussion wurde die Sitzung für beendet erklärt.
After a long discussion, the meeting was declared over.
The verb consists of the prefix 'er-' and the verb 'klären' (to clarify/make clear). The prefix 'er-' often expresses the achievement of a goal or result. 'Erklären' (to explain/declare) therefore means creating the result of clarity.
Können Sie diesen Punkt bitte noch einmal klären?
Could you please clarify this point once more?
'Erklären' (to explain/declare) is an inseparable verb. The prefix 'er-' remains firmly attached to the verb stem in all tenses and sentence constructions.
Er erklärte mir geduldig den Weg.
He patiently explained the way to me.
When you explain something to someone, the person is in the dative and the thing is in the accusative. The pattern is: 'jemandem (dative) etwas (accusative) erklären' (to explain something to someone).
Ich erkläre dir die Regeln des Spiels.
I am explaining the rules of the game to you.
This meaning is used in formal and official contexts such as politics, law, or business. The object is often an abstract noun, e.g., 'den Krieg erklären' (to declare war), 'seine Absicht erklären' (to declare one's intention), or 'den Notstand erklären' (to declare a state of emergency).
Das Land erklärte seinem Nachbarn den Krieg.
The country declared war on its neighbor.
In the reflexive form 'sich erklären' (to be explained), the logical puzzle or phenomenon is the subject of the sentence. The reason is often introduced with the preposition 'durch' (by/through).
Sein seltsames Verhalten erklärt sich durch seine Müdigkeit.
His strange behavior is explained by his tiredness.
In this meaning, the preposition 'für' is always used, followed by an adjective or participle. The structure is: 'jemanden/etwas für ... erklären' (to declare someone/something ...).
Der Richter erklärte den Angeklagten für schuldig.
The judge declared the defendant guilty.
to justify / establish
Erklären and begründen both make a connection understandable and can both be used when discussing a decision. Erklären can present the process or cause objectively, while begründen explicitly gives the reasons for a statement or action. Use erklären for a neutral explanation of a cause, and begründen more for a justification.

Die Ärztin erklärt, warum die Behandlung heute länger dauert.
The doctor explains why the treatment is taking longer today.
Die Ärztin begründet, warum sie diese Behandlung heute empfiehlt.
The doctor justifies why she recommends this treatment today.