Why and how does the discriminant rule work?!?

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My physics teacher asked us how the formula works, thanks for this wonderful explanation

Wakinss
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the quadratic formula is all over 2a not 2c no???

Asheyboi
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I was wondering more about why it is presicely b²-4ac, and why not some other cobination of terms that needs to be concidered

Rulian_Sama