数学 - Python - 代数学 - 1次方程式, 2次方程式 - 解の公式、解の虚実 - 2次方程式の解の公式、複素数の範囲

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代数への出発 (新装版数学入門シリーズ) (松坂和夫(著)、岩波書店)の第4章(1次方程式, 2次方程式 )、4(解の公式、解の虚実)、2次方程式の解の公式の問18の解答を求めてみる。

1. $\begin{array}{l} x^{2} = - 8 \\ x = \pm \sqrt{- 8} \\ = \pm 2 \sqrt{2} i \end{array}$
2. $\begin{array}{l} x^{2} = - \frac{9}{16} \\ x = \pm \frac{3}{4} i \end{array}$
3. $\begin{array}{l} x^{2} = - \frac{27}{4} \\ x = \pm \frac{3 \sqrt{3}}{2} i \end{array}$
4. $\begin{array}{l} x \\ = \frac{- 1 \pm \sqrt{1 - 4}}{2} \\ = \frac{- 1 \pm \sqrt{3} i}{2} \end{array}$
5. $\begin{array}{l} x \\ = \frac{- 3 \pm \sqrt{9 - 32}}{8} \\ = \frac{- 3 \pm \sqrt{23} i}{8} \end{array}$
6. $\begin{array}{l} x \\ = 1 \pm \sqrt{1 - 5} \\ = 1 \pm 2 i \end{array}$
7. $\begin{array}{l} x \\ = \frac{- 4 \pm \sqrt{16 - 18}}{2} \\ = \frac{- 4 \pm \sqrt{2} i}{2} \end{array}$
8. $\begin{array}{l} x \\ = \frac{6 \pm \sqrt{36 - 63}}{9} \\ = \frac{6 \pm \sqrt{27} i}{9} \\ = \frac{2 \pm \sqrt{3} i}{3} \end{array}$

#!/usr/bin/env python3 from unittest import TestCase, main from sympy import symbols, sqrt, I, Rational, solveset, plot, pprint print('18.') x = symbols('x', imag=True) fs = [x ** 2 + 8, 16 * x ** 2 + 9, 4 * x ** 2 + 27, x ** 2 + x + 1, 4 * x ** 2 + 3 * x + 2, x ** 2 - 2 * x + 5, 2 * x ** 2 + 8 * x + 9, 9 * x ** 2 - 12 * x + 7, ] sign = [-1, 1] class MyTestCase(TestCase): def test(self): zss = [{a + s * b * I for s in sign} for a, b in [(0, 2 * sqrt(2)), (0, Rational(3, 4)), (0, 3 * sqrt(3) / 2), (-Rational(1, 2), sqrt(3) / 2), (-Rational(3, 8), sqrt(23) / 8), (1, 2), (-2, sqrt(2) / 2), (Rational(2, 3), sqrt(3) / 3)]] for i, (f, zs) in enumerate(zip(fs, zss), 1): print(f'({i})') self.assertEqual(solveset(f), zs) p = plot(*fs, ylim=(0, 20), legend=False, show=False) colors = ['red', 'green', 'blue', 'brown', 'orange', 'purple', 'pink', 'gray', 'skyblue', 'yellow'] for o, color in zip(p, colors): o.line_color = color for o in zip(fs, colors): pprint(o) p.show() p.save(f'sample18.png') if __name__ == "__main__": main()

% ./sample18.py -v 18. ⎛ 2 ⎞ ⎝x + 8, red⎠ ⎛ 2 ⎞ ⎝16⋅x + 9, green⎠ ⎛ 2 ⎞ ⎝4⋅x + 27, blue⎠ ⎛ 2 ⎞ ⎝x + x + 1, brown⎠ ⎛ 2 ⎞ ⎝4⋅x + 3⋅x + 2, orange⎠ ⎛ 2 ⎞ ⎝x - 2⋅x + 5, purple⎠ ⎛ 2 ⎞ ⎝2⋅x + 8⋅x + 9, pink⎠ ⎛ 2 ⎞ ⎝9⋅x - 12⋅x + 7, gray⎠ test (__main__.MyTestCase) ... (1) (2) (3) (4) (5) (6) (7) (8) ok ---------------------------------------------------------------------- Ran 1 test in 0.228s OK %

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2020年2月18日火曜日

数学 - Python - 代数学 - 1次方程式, 2次方程式 - 解の公式、解の虚実 - 2次方程式の解の公式、複素数の範囲

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