<prompt> Zingatia chaguo za kukokotoa zx,y zinazoelezea paraboloidz 2x - y2 - 2y2 - 3y.Archimedes na Brahmagupta wanacheza mchezo. Archimedes kwanza huchagua x. Baadaye, Brahmagupta anachagua y. Archimedes angependa kupunguza z huku Brahmagupta akitaka kuongeza z. Kwa kuchukulia kwamba Brahmagupta itacheza vyema, ni thamani gani ya x ambayo Archimedes anapaswa kuchagua?  <response>: Kupanua z, tunapataanzaalignz 4x2 - 4xy y2 - 2y2 - 3y -y2 - 4x 3 y 4x2.endalignBaada ya Archimedes kuchagua x, Brahmagupta atachaguay -frac4x 32ili kuongeza z. Kishaanzaalignz -kushoto -frac4x 32 kulia2 - 4x 3 kushoto -frac4x 32 kulia2 4x 2 8x2 6x frac94.endalignIli kupunguza usemi huu, Archimedes wanapaswa kuchagua x -frac616 -frac38. 