<prompt> Bainisha fx frac1x frac1xna gx frac1x - frac1x .Tafuta mraba wa suluhu halisi kubwa zaidi la mlingano gx2 - fx2 tfrac54.  <response>: Kwanza tunarahisisha misemo ya fx na gx. Tuna fx frac1x frac1x frac1 tfrac x21x fracxx21 na gx frac1x-frac1x frac1 tfracx2-1x fracxx2-1.Kwa hivyo, anzailiyopangwa gx2 - f x2 fracx2x2-12 - fracx2x212 fracx2leftx212-x2-12kuliax2-12x212 fracx24x2x4-12 frac4x4x4-12 . endalignedKwa hivyo, tuna mlingano frac4x4x4-12 frac54.Kuzidisha na kupanua mtambuka , tunapata 16x4 5x8 - 10x4 5,hivyo 0 5x8 - 26x4 5.Vigezo hivi kama 0 5x4- 1x4-5,kwa hivyo ama x4 tfrac15 au x4 5. Inafuata kwamba suluhisho kubwa zaidi la mlinganyo ni x sqrt4 5, hivyo x2 sqrt5. 