Решебник по алгебре и начала математического анализа 11 класс Колягин Задание 649

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

\[= (z - 2 + i)(z - 2 - i)\]

\[D = 16 - 20 = - 4\]

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

\[2)\ z^{2} + 4z + 13 =\]

\[= (z + 2 + 3i)(z + 2 - 3i)\]

\[D = 16 - 52 = - 36\]

\[z = \frac{- 4 \pm \sqrt{- 36}}{2} = \frac{- 4 \pm 6i}{2} =\]

\[= - 2 \pm 3i.\]

\[3)\ z^{2} + 2z + 4 =\]

\[= \left( z + 1 + \sqrt{3}i \right)\left( z + 1 - \sqrt{3}i \right)\]

\[D = 4 - 16 = - 12\]

\[z = \frac{- 2 \pm \sqrt{- 12}}{2} = \frac{- 2 \pm 2\sqrt{3}i}{2} =\]

\[= - 1 \pm \sqrt{3}i.\]

\[4)\ z^{2} - 6z + 11 =\]

\[= \left( z - 3 + \sqrt{2}i \right)\left( z - 3 - \sqrt{2}i \right)\]

\[D = 36 - 44 = - 8\]

\[z = \frac{6 \pm \sqrt{- 8}}{2} = \frac{6 \pm 2\sqrt{2}i}{2} =\]

\[= 3 \pm \sqrt{2}i.\]