Решите уравнение:
[tex] \sqrt{ \frac{x}{x+1} } + \sqrt{ \frac{x+1}{x} } = \frac{5}{2} [/tex]
[tex] \sqrt{ \frac{x}{x+1} } + \sqrt{ \frac{x+1}{x} } = \frac{5}{2} [/tex]
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(√(х /(х+1)) +√((х+1) /х))² =(5/2)²
√(х /(х+1))²+2*√(х /(х+1))*√((х+1) /х)+ √((х+1) /х)² =25/4
х /(х+1) + 2*√((х /(х+1))*((х+1) /х))+ (х+1) /х =25/4
х /(х+1) + 2 + (х+1) /х =25/4
х*х +(х+1)*(х+1)
---------------------- = 4,25
х*(х+1)
х*х +(х+1)*(х+1) =4.25*х*(х+1)
х²+х²+2х+1=4,25х²+4,25х
2.25х²+2.25х-1=0
D=5,0625+9=14,0625 √D=3,75
x₁=(-2,25+3,75)/4,5=1,5/4,5 =1/3
x₂=(-2,25-3,75)/4,5= -6/4,5= - 4/3