[tex]Zad. 1\\y=-2x^2+5x+1\\\Delta=5^2-4*(-2)*1=25+8=33\\\sqrt{\Delta}=\sqrt{33}\\x_1=\frac{-5-\sqrt{33}}{-4}=\frac{-(5+\sqrt{33})}{-4}=\frac{5+\sqrt{33}}4\\x_2=\frac{-5+\sqrt{33}}{-4}=\frac{-(5-\sqrt{33})}{-4}=\frac{5-\sqrt{33}}4[/tex]
[tex]x_1^4=((\frac{5+\sqrt{33}}4)^2)^2=(\frac{25+10\sqrt{33}+33}{16})^2=(\frac{58+10\sqrt{33}}{16})^2=(\frac{29+5\sqrt{33}}{8})^2=\frac{841+290\sqrt{33}+825}{64}=\frac{833+145\sqrt{33}}{32}[/tex]
[tex]x_2^4=((\frac{5-\sqrt{33}}4)^2)^2=(\frac{25-10\sqrt{33}+33}{16})^2=(\frac{29-5\sqrt{33}}{8})^2=\frac{841-290\sqrt{33}+825}{64}=\frac{833-145\sqrt{33}}{32}[/tex]
[tex]x_1^4+x_2^4=\frac{833+145\sqrt{33}}{32}+\frac{833-145\sqrt{33}}{32}=\frac{2*833}{32}=\frac{833}{16}[/tex]
[tex]Zad. 2[/tex]
[tex]x_1+x_2=-3\\\frac{1}{x_1}+\frac{1}{x_2}=21\\\frac{x_2}{x_1x_2}+\frac{x_1}{x_1x_2}=21\\\frac{x_1+x_2}{x_1x_2}=21 /*x_1x_2\\x_1+x_2=21x_1x_2\\-3=21x_1x_2 /:21\\-\frac17=x_1x_2[/tex]
[tex]x_1x_2=\frac{c}a\\\frac{c}a=-\frac17\\c=-\frac17a\\\\x_1+x_2=\frac{-b}a\\\frac{-b}a=-3\\-b=-3a\\b=3a\\\\ax^2+bx+c=0\\ax^2+3ax-\frac17a=0 /:a\\a(x^2+3x-\frac17)=0 /:a\\\\\underline{x^2+3x-\frac17=0}[/tex]
