Rozwiąż równanie [latex]log_{ frac{1}{2}x+1 }4x=2 [/latex]

[latex]oxed{Log_{a}b=c implies a^{c}=b} \ \ \ \ D: frac{1}{2}x+1 extgreater 0 wedge frac{1}{2}x+1 eq 1 wege 4x extgreater 0 \ \ frac{1}{2}x extgreater -1 wedge frac{1}{2}x eq 0 wedge x extgreater 0 \ \ x extgreater 0 \ \ log_{ frac{1}{2}x+1} 4x = 2 \ \ ( frac{1}{2}x+1)^{2}=4x \ \ frac{1}{4}x^{2} + x +1 = 4x /*4 \ \ x^{2}+4x+4=16x \ \ x^{2} - 12x + 4 =0 \ \ Delta = (-12)^{2}-4*1*4 = 144 - 16 =128[/latex] [latex] sqrt{Delta} = 8 sqrt{2} \ \ x_{1} = frac{12-8 sqrt{2} }{2}=6 - 4 sqrt{2} \ \ x_{2}= frac{12+8sqrt{2} }{2} =6+4 sqrt{2} [/latex]