Решить уравнение x^2+1/x^2+3(x+1/x)=8
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![x+frac{1}{x}=t\
x^2+frac{1}{x}^2=t^2-2\\
t^2-2+3t=8\
t^2+3t-10=0\
D=9+4*1*10=7^2\
t=frac{-3+7}{2}=2\
t=frac{-3-7}{2}=-5\
\
x+frac{1}{x}=2\
x^2-2x+1=0\
(x-1)^2=0\
x=1\\
x^2+5x+1=0\
D=25-4*1*1=sqrt{21}\
x=frac{-5+sqrt{21}}{2}\
x=frac{-5-sqrt{21}}{2}](https://tex.z-dn.net/?f=+x%2Bfrac%7B1%7D%7Bx%7D%3Dt%5C%0Ax%5E2%2Bfrac%7B1%7D%7Bx%7D%5E2%3Dt%5E2-2%5C%5C%0A+t%5E2-2%2B3t%3D8%5C%0A+t%5E2%2B3t-10%3D0%5C%0A+D%3D9%2B4%2A1%2A10%3D7%5E2%5C%0A+t%3Dfrac%7B-3%2B7%7D%7B2%7D%3D2%5C%0A+t%3Dfrac%7B-3-7%7D%7B2%7D%3D-5%5C%0A%5C%0A+x%2Bfrac%7B1%7D%7Bx%7D%3D2%5C%0A+x%5E2-2x%2B1%3D0%5C%0A+%28x-1%29%5E2%3D0%5C%0A+x%3D1%5C%5C%0A+x%5E2%2B5x%2B1%3D0%5C%0A+D%3D25-4%2A1%2A1%3Dsqrt%7B21%7D%5C%0A+x%3Dfrac%7B-5%2Bsqrt%7B21%7D%7D%7B2%7D%5C%0A+x%3Dfrac%7B-5-sqrt%7B21%7D%7D%7B2%7D "x+frac{1}{x}=t\
x^2+frac{1}{x}^2=t^2-2\\
t^2-2+3t=8\
t^2+3t-10=0\
D=9+4*1*10=7^2\
t=frac{-3+7}{2}=2\
t=frac{-3-7}{2}=-5\
\
x+frac{1}{x}=2\
x^2-2x+1=0\
(x-1)^2=0\
x=1\\
x^2+5x+1=0\
D=25-4*1*1=sqrt{21}\
x=frac{-5+sqrt{21}}{2}\
x=frac{-5-sqrt{21}}{2}")
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