[tex]lg^2(x) - lg(x^2) + 1 = 6\lg^2(x) - 2lg(x) + 1 = 6\(lg(x) - 1)^2 = 6\(lg(x) - 1 - sqrt{6})(lg(x) - 1 + sqrt{6}) = 0\x_1 = 10^{1 + sqrt{6}}\x_2 = 10^{1 - sqrt{6}}[/tex]

[tex]lg^{2}x-lgx^{2}+1=6\\lg^{2} x-2lgx-5=0\\lgx=m,x>0\\m^{2}-2m-5=0\\D=(-2)^{2} -4*(-5)=4+20=24=(2sqrt{6})^{2}\\m_{1}=frac{2+2sqrt{6} }{2}=1+sqrt{6}\\m_{2}=frac{2-2sqrt{6} }{2}=1-sqrt{6}\\lgx=1+sqrt{6} \\x_{1}=10^{1+sqrt{6} }\\lgx=1-sqrt{6}\\x_{2}=10^{1-sqrt{6} }[/tex]