What is the right way to pronounce большую?
Is this a special case that you just have to live with?
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Большая (with the emphasis on the "а") — is the adjective "big" for feminine nouns, like большая победа (a big victory, a big win), большая машина (a big car) etc. большую in this case is the word большая in the genitive or accusative case.
Большая (with the emphasis on the "о") — is a bit complicated. Roughly speaking, it's an equivalent of the word "bigger" in English, but with considerable catches:
Which car should I buy? Take a bigger one. -> Какую машину стоит мне купить? Возьми большую.
This car is bigger, than that one. -> Эта машина больше той
NOT Эта машина большая той
The word большую in case 2 is the word большая in the genitive or instrumental case, likewise in case 1.
So to summarize, both ways are correct, because the words are different.
To summarize all the cases of this kind (большая — большая), here is the list (the adjectives here also have to be declined properly, this is only their initial form in the nominative case):
A compact explanation. These are two words of different meaning and stress position which doesn't vary with cases. Depending on case/gender, the two words may differ in spelling:
Большо́й (-а́я, fem.) means big.
Бо́льший (-ая, fem.) means the bigger (one/part - of whatever is available).
Бо́льшую is the accusative case (fem. gender, singular) of бо́льшая and большу́ю is the accusative case (fem. gender, singular) of больша́я (the spelling in the accusative/nominative coincides for feminine). The spelling here (fem. + singular) will differ for cases other than nominative/accusative: большо́й/бо́льшей.
As you can see, the stress position always corresponds to the initial adjective's meaning.