wzory Viete’a:
x_1+x_2=-\frac{b}{a}
x_1*x_2=\frac{c}{a}
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f(x)=2x^2+4x-6
a=2, b=4, c=-6
a)
{x_1}^2+{x_2}^2=(x_1+x_2)^2-2x_1x_2=(-\frac{b}{a})^2-2*\frac{c}{a}=\frac{b^2}{a^2}-\frac{2c}{a}=\frac{b^2-2ac}{a^2}=\frac{4^2-2*2*(-6)}{2^2}=\frac{16+24}{4}=\frac{40}{4}=10
b)
\frac{1}{x_1}+\frac{1}{x_2}=\frac{x_2+x_1}{x_1x_2}=\frac{-\frac{b}{a}}{\frac{c}{a}}=-\frac{b}{a}*\frac{a}{c}=\frac{-b}{c}=\frac{-4}{-6}=\frac{2}{3}
c)
\frac{1}{{x_1}^2}+\frac{1}{{x_2}^2}=\frac{{x_2}^2+{x_1}^2}{{x_1}^2{x_2}^2}=\frac{(x_1+x_2)^2-2x_1x_2}{(x_1x_2)^2}=\frac{(-\frac{b}{a})^2-2*\frac{c}{a}}{(\frac{c}{a})^2}=
=\frac{\frac{b^2}{a^2}-\frac{2c}{a}}{\frac{c^2}{a^2}}=\frac{b^2-2ac}{a^2}*\frac{a^2}{c^2}=\frac{b^2-2ac}{c^2}=
=\frac{4^2-2*2*(-6)}{(-6)^2}=\frac{16+24}{36}=\frac{40}{36}=\frac{10}{9}=1\frac{1}{9}
d)
(x_1-x_2)^2={x_1}^2-2x_1x_2+{x_2}^2={x_1}^2+{x_2}^2-2x_1x_2=(x_1+x_2)^2-2x_1x_2-2x_1x_2=
=(-\frac{b}{a})^2-4x_1x_2=\frac{b^2}{a^2}-4*\frac{c}{a}=\frac{b^2-4ac}{a^2}=\frac{4^2-4*2*(-6)}{2^2}=\frac{16+48}{4}=\frac{64}{4}=16
e)
\frac{x_2}{x_1}+\frac{x_1}{x_2}=\frac{{x_2}^2+{x_1}^2}{x_1x_2}=\frac{(x_1+x_2)^2-2x_1x_2}{x_1x_2}=\frac{(-\frac{b}{a})^2}{x_1x_2}-2=\frac{\frac{b^2}{a^2}}{\frac{c}{a}}-2=
=\frac{b^2}{a^2}*\frac{a}{c}-2=\frac{b^2}{ac}-2=\frac{4^2}{2*(-6)}-2=\frac{16}{-12}-2=-\frac{4}{3}-2=-1\frac{1}{3}-2=-3\frac{1}{3}