x²-5x+1=0
Δ=(-5)²-4·1·1=25-4=21
[tex]x_{1, \ 2}=\frac{-b\pm \sqrt{\Delta}}{2a}\\ \\ x_1=\frac{5+\sqrt{21}}{2}\\ \\ x_2=\frac{5-\sqrt{21}}{2}[/tex]
x | -∞ 0 [tex]\frac{5-\sqrt{21}}{2}[/tex] [tex]\frac{5+\sqrt{21}}{2}[/tex] +∞
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x | - - - - - -0 + + + + + + + + + + + + + + + + + +
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x²-5x+1 | - - - - - - - - - - - - - - 0 + + + + + 0 - - - - - - - -
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[tex]\frac{x^2-5x+1}{x}[/tex] | + + + + | - - - - - - - 0 + + + + + 0 - - - - - - - -
[tex]\frac{x^2-5x+1}{x} \leq 0 \Leftrightarrow x \in (0; \ \frac{5-\sqrt{21}}{2}] \ U \ [\frac{5+\sqrt{21}}{2}; \ +oo)[/tex]
