数学 – Python - JavaScript - 数学はここから始まる - 数 – 2次方程式と複素数 – 2数を解とする方程式(対称式)

数学読本〈1〉
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学習環境

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• Windows 10 Pro (OS)
• Nebo(Windows アプリ)
• iPad Pro + Apple Pencil
• MyScript Nebo - MyScript(iPad アプリ(iOS))
• 参考書籍
  • Pythonからはじめる数学入門 (楽天ブックス(Kobo) Yahoo!)
  • JavaScriptリファレンス 第6版 (楽天ブックス(Kobo) Yahoo!)
  • インタラクティブ・データビジュアライゼーション (楽天ブックス(Kobo) Yahoo!)

数学読本〈1〉数・式の計算/方程式/不等式 (松坂 和夫(著)、岩波書店)の第3章(数学の威力を発揮する - 方程式)、3.2(2次方程式と複素数)、2数を解とする方程式の問14.を取り組んでみる。

14. 
    
    1. 
       
       $\begin{array}{}\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}=-\frac{4}{2}=2\\ \mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}=\frac{5}{2}\\ \left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+3\right)+\left(\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}+3\right)\\ =\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}+6\\ =2+6\\ =8\\ \left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+3\right)\left(\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}+3\right)\\ =\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}+9+3\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)\\ =\frac{5}{2}+9+6\\ =\frac{35}{2}\\ 2x^2-16x+35=0\end{array}$
    2. 
       
       $\begin{array}{}{\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^2+{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^2\\ ={\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)}^2-2\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\\ =4-5\\ =-1\\ {\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}}^2{\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}}^2=\frac{25}{4}\\ 4x^2+4x+25=0\end{array}$
    3. 
       
       $\begin{array}{}\frac{1}{2\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}-1}+\frac{1}{2\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}-1}\\ =\frac{2\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)-2}{4\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}+1-2\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)}\\ =\frac{4-2}{10+1-4}\\ =\frac{2}{7}\\ \frac{1}{2\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}-1}·\; <!--\; middle\; dot\; -->\frac{1}{2\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}-1}\\ =\frac{1}{4\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}-2\left(\mathrm{\alpha \; <!--\; greek\; small\; letter\; alpha\; -->}+\mathrm{\beta \; <!--\; greek\; small\; letter\; beta\; -->}\right)+1}\\ =\frac{1}{10-4+1}\\ =\frac{1}{7}\\ 7x^2-2x+1=0\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, solve, plot

print('14.')

x = symbols('x')
alpha, beta = solve(2 * x ** 2 - 4 * x + 5, x)
ts = [(2, alpha + 3, beta + 3),
      (4, alpha ** 2, beta ** 2),
      (7, 1 / (2 * alpha - 1), 1 / (2 * beta - 1))]

for i, (a, alpha0, beta0) in enumerate(ts, 1):
    print(f'({i})')
    pprint((a * (x - alpha0) * (x - beta0)).simplify())

入出力結果(Terminal, Jupyter(IPython))

$ ./sample14.py
14.
(1)
   2            
2⋅x  - 16⋅x + 35
(2)
   2           
4⋅x  + 4⋅x + 25
(3)
   2          
7⋅x  - 2⋅x + 1
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="1">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.001" value="0.01">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-25">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="25">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-25">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="25">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample14.js"></script>

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],
    p = (x) => pre0.textContent += x + '\n';

let fns = [[x => 2 * x ** 2 - 16 * x + 35, 'red'],
           [x => 4 * x ** 2 + 4 * x + 25, 'green'],
           [x => 7 * x ** 2 - 2 * x + 1, 'blue']];
    
let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', (d) => d[4] || 'black');

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d) => d[2] || 'green');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);

    [fns].forEach((fs) => p(fs.join('\n')));
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =