$x_1+x_2=\frac{-b}{a}$  

$x_1\cdot x_2=\frac{c}{a}$  

$\frac{1}{x_1}+\frac{1}{x_2}=\frac{x_2+x_1}{x_1{x}_2}=\frac{\frac{-b}{a}}{\frac{c}{a}}=\frac{-b}{a}\cdot \frac{a}{c}=\frac{-b}{c}$  

$x_1^2+x_2^2={\left(x_1+x_2\right)}^2-2x_1{x}_2={\left(\frac{-b}{a}\right)}^2-2\cdot \frac{c}{a}$  

$\frac{1}{x_1^2}+\frac{1}{x_2^2}=\frac{x_2^2}{x_1^2{x}_2^2}+\frac{x_1^2}{x_1^2{x}_2^2}=\frac{x_1^2+x_2^2}{{\left(x_1{x}_2\right)}^2}=\frac{\frac{b^2}{a^2}-2\frac{c}{a}}{{\left(\frac{c}{a}\right)}^2}=\frac{\frac{b^2-2ac}{a^2}}{\frac{c^2}{a^2}}=\frac{b^2-2ac}{a^2}\cdot \frac{a^2}{c^2}=\frac{b^2-2ac}{c^2}$  

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Równanie	$x_1+x_2$	$x_1\cdot x_2$	$\frac{1}{x_1}+\frac{1}{x_2}$	$x_1^2+x_2^2$	$\frac{1}{x_1^2}+\frac{1}{x_2^2}$
$4x^2-20x+5=0$	$5$	$1\frac{1}{4}$	$4$	$22\frac{1}{2}$	$14\frac{2}{5}$
$2x^2+3x-3=0$	$-1\frac{1}{2}$	$-1\frac{1}{2}$	$1$	$5\frac{1}{4}$	$2\frac{1}{3}$
$9x^2-15x-19=0$	$1\frac{2}{3}$	$-2\frac{1}{9}$	$-\frac{15}{19}$	$7$	$1\frac{206}{361}$
$3x^2-6x+2=0$	$2$	$\frac{2}{3}$	$3$	$2\frac{2}{3}$	$6$

Odpowiedź: ALGEBRA

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Obliczenia dla pierwszego wiersza:

$4x^2-20x+5=0$  

$x_1+x_2=\frac{20}{4}=5$  

$x_1\cdot x_2=\frac{5}{4}=1\frac{1}{4}$  

$\frac{1}{x_1}+\frac{1}{x_2}=\frac{20}{5}=4$  

$x_1^2+x_2^2={\left(\frac{20}{4}\right)}^2-2\cdot \frac{5}{4}=25-\frac{10}{4}=25-2\frac{1}{2}=22\frac{1}{2}$  

$\frac{1}{x_1^2}+\frac{1}{x_2^2}=\frac{{\left(-20\right)}^2-2\cdot 4\cdot 5}{5^2}=\frac{400-40}{25}=\frac{360}{25}=14\frac{10}{25}=14\frac{2}{5}$  

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Obliczenia dla drugiego wiersza:

$2x^2+3x-3=0$  

$x_1+x_2=\frac{-3}{2}=-1\frac{1}{2}$  

$x_1\cdot x_2=\frac{-3}{2}=-1\frac{1}{2}$  

$\frac{1}{x_1}+\frac{1}{x_2}=\frac{-3}{-3}=1$  

$x_1^2+x_2^2={\left(\frac{-3}{2}\right)}^2-2\cdot \left(\frac{-3}{2}\right)=\frac{9}{4}+\frac{6}{2}=\frac{9}{4}+\frac{12}{4}=\frac{21}{4}=5\frac{1}{4}$  

$\frac{1}{x_1^2}+\frac{1}{x_2^2}=\frac{3^2-2\cdot 2\cdot \left(-3\right)}{{\left(-3\right)}^2}=\frac{9+12}{9}=\frac{21}{9}=2\frac{3}{9}=2\frac{1}{3}$  

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Obliczenia dla trzeciego wiersza: 

$9x^2-15x-19=0$  

$x_1+x_2=\frac{15}{9}=\frac{5}{3}=1\frac{2}{3}$  

$x_1\cdot x_2=\frac{-19}{9}=-2\frac{1}{9}$  

$\frac{1}{x_1}+\frac{1}{x_2}=\frac{15}{-19}=-\frac{15}{19}$  

$x_1^2+x_2^2={\left(\frac{15}{9}\right)}^2-2\cdot \frac{-19}{9}={\left(\frac{5}{3}\right)}^2+\frac{38}{9}=\frac{25}{9}+\frac{38}{9}=\frac{63}{9}=7$  

$\frac{1}{x_1^2}+\frac{1}{x_2^2}=\frac{{\left(-15\right)}^2-2\cdot 9\cdot \left(-19\right)}{{\left(-19\right)}^2}=\frac{225+342}{361}=\frac{567}{361}=1\frac{206}{361}$  

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Obliczenia dla czwartego wiersza:

$3x^2-6x+2=0$  

$x_1+x_2=\frac{6}{3}=2$  

$x_1\cdot x_2=\frac{2}{3}$  

$\frac{1}{x_1}+\frac{1}{x_2}=\frac{6}{2}=3$  

$x_1^2+x_2^2={\left(\frac{6}{3}\right)}^2-2\cdot \frac{2}{3}=2^2-\frac{4}{3}=4-1\frac{1}{3}=2\frac{2}{3}$  

$\frac{1}{x_1^2}+\frac{1}{x_2^2}=\frac{{\left(-6\right)}^2-2\cdot 3\cdot 2}{2^2}=\frac{36-12}{4}=\frac{24}{4}=6$  

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