Решение квадратных уравнений через комплексные числа

Для решения данных уравнений на множестве комплексных чисел будем использовать формулу корней квадратного уравнения: \[x = \frac{-b \pm \sqrt{D}}{2a}\] где \(D = b^2 - 4ac\). Если \(D < 0\), то \(\sqrt{D} = \sqrt{|D|} \cdot i\), где \(i\) — мнимая единица (\(i^2 = -1\)). 1) \(3x^2 + 12x + 15 = 0\) Разделим на 3: \(x^2 + 4x + 5 = 0\) \(D = 4^2 - 4 \cdot 5 = 16 - 20 = -4\) \(x = \frac{-4 \pm \sqrt{-4}}{2} = \frac{-4 \pm 2i}{2} = -2 \pm i\) 2) \(x^2 - 6x + 13 = 0\) \(D = (-6)^2 - 4 \cdot 13 = 36 - 52 = -16\) \(x = \frac{6 \pm \sqrt{-16}}{2} = \frac{6 \pm 4i}{2} = 3 \pm 2i\) 3) \(4x^2 + 4x + 17 = 0\) \(D = 4^2 - 4 \cdot 4 \cdot 17 = 16 - 272 = -256\) \(x = \frac{-4 \pm \sqrt{-256}}{8} = \frac{-4 \pm 16i}{8} = -0,5 \pm 2i\) 4) \(x^2 + 10x + 34 = 0\) \(D = 10^2 - 4 \cdot 34 = 100 - 136 = -36\) \(x = \frac{-10 \pm \sqrt{-36}}{2} = \frac{-10 \pm 6i}{2} = -5 \pm 3i\) 5) \(x^2 + 3x + 4 = 0\) \(D = 3^2 - 4 \cdot 4 = 9 - 16 = -7\) \(x = \frac{-3 \pm \sqrt{-7}}{2} = \frac{-3 \pm i\sqrt{7}}{2} = -1,5 \pm \frac{\sqrt{7}}{2}i\) 6) \(2x^2 + 8x + 11 = 0\) \(D = 8^2 - 4 \cdot 2 \cdot 11 = 64 - 88 = -24\) \(x = \frac{-8 \pm \sqrt{-24}}{4} = \frac{-8 \pm 2i\sqrt{6}}{4} = -2 \pm \frac{\sqrt{6}}{2}i\) 7) \(x^2 + 16 = 0\) \(x^2 = -16\) \(x = \pm \sqrt{-16} = \pm 4i\) 8) \(3x^2 - 6x + 6 = 0\) Разделим на 3: \(x^2 - 2x + 2 = 0\) \(D = (-2)^2 - 4 \cdot 2 = 4 - 8 = -4\) \(x = \frac{2 \pm \sqrt{-4}}{2} = \frac{2 \pm 2i}{2} = 1 \pm i\) 9) \(x^2 + 5x + 8 = 0\) \(D = 5^2 - 4 \cdot 8 = 25 - 32 = -7\) \(x = \frac{-5 \pm \sqrt{-7}}{2} = -2,5 \pm \frac{\sqrt{7}}{2}i\) 10) \(4x^2 - 12x + 18 = 0\) Разделим на 2: \(2x^2 - 6x + 9 = 0\) \(D = (-6)^2 - 4 \cdot 2 \cdot 9 = 36 - 72 = -36\) \(x = \frac{6 \pm \sqrt{-36}}{4} = \frac{6 \pm 6i}{4} = 1,5 \pm 1,5i\) 11) \(x^2 + x + 1 = 0\) \(D = 1^2 - 4 = -3\) \(x = \frac{-1 \pm \sqrt{-3}}{2} = -0,5 \pm \frac{\sqrt{3}}{2}i\) 12) \(2x^2 + 10x + 27 = 0\) \(D = 10^2 - 4 \cdot 2 \cdot 27 = 100 - 216 = -116\) \(x = \frac{-10 \pm \sqrt{-116}}{4} = \frac{-10 \pm 2i\sqrt{29}}{4} = -2,5 \pm \frac{\sqrt{29}}{2}i\) 13) \(x^2 - 8x + 20 = 0\) \(D = (-8)^2 - 4 \cdot 20 = 64 - 80 = -16\) \(x = \frac{8 \pm \sqrt{-16}}{2} = \frac{8 \pm 4i}{2} = 4 \pm 2i\)