Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 650

\[1)\ 4z^{2} + 4z + 5 =\]

\[= 4\left( z + \frac{1}{2} + i \right)\left( z + \frac{1}{2} - i \right)\]

\[D = 16 - 80 = - 64\]

\[z = \frac{- 4 \pm \sqrt{- 64}}{2 \bullet 4} = \frac{- 4 \pm 8i}{8} =\]

\[= - \frac{1}{2} \pm i.\]

\[2)\ 16z^{2} - 32z + 17 =\]

\[= 16\left( z - 1 + \frac{1}{4}i \right)\left( z - 1 - \frac{1}{4}i \right)\]

\[D = 1024 - 1088 = - 64\]

\[z = \frac{32 \pm \sqrt{- 64}}{2 \bullet 16} = \frac{32 \pm 8i}{32} =\]

\[= 1 \pm \frac{1}{4}i.\]

\[3)\ 25z^{2} + 50z + 26 =\]

\[= 25\left( z + 1 + \frac{1}{5}i \right)\left( z + 1 - \frac{1}{5}i \right)\]

\[D = 2500 - 2600 = - 100\]

\[z = \frac{- 50 \pm \sqrt{- 100}}{2 \bullet 25} =\]

\[= \frac{- 50 \pm 10i}{50} = - 1 \pm \frac{1}{5}i.\]

\[4) - z^{2} + 10z - 26 =\]

\[= - (z - 5 + i)(z - 5 - i)\]

\[D = 100 - 104 = - 4\]

\[z = \frac{10 \pm \sqrt{- 4}}{2} = \frac{10 \pm 2i}{2} = 5 \pm i.\]