数学 – Python - JavaScript - 数学はここから始まる - 数 – 高次方程式 – 連立2次方程式(消去法、2次方程式の解と係数の関係)

数学読本〈1〉
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• 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)、3.3(高次方程式)、連立2次方程式の問32.を取り組んでみる。

32. 
    
    1. 
       
       $\begin{array}{}y=x-1\\ x^2+{\left(x-1\right)}^2=25\\ 2x^2-2x-24=0\\ x^2-x-12=0\\ \left(x-4\right)\left(x+3\right)=0\\ x=-3,y=-4\\ x=4,y=3\end{array}$
    2. 
       
       $\begin{array}{}{x}^2+3x^2=48\\ x^2=12\\ x=\pm \; <!--\; plus-minus\; sign\; -->2\sqrt{3}\\ x=\pm \; <!--\; plus-minus\; sign\; -->2\sqrt{3},y=\pm \; <!--\; plus-minus\; sign\; -->6\end{array}$
    3. 
       
       $\begin{array}{}{\left(2x-1\right)}^2-x^2=5\\ 4x^2-4x+1-x^2-5=0\\ 3x^2-4x-4=0\\ \left(x-2\right)\left(3x+2\right)=0\\ x=-\frac{2}{3},y=-\frac{4}{3}-1=-\frac{7}{3}\\ x=2,y=4-1=3\end{array}$
    4. 
       
       $\begin{array}{}{t}^2-4t+2=0\\ t=2\pm \; <!--\; plus-minus\; sign\; -->\sqrt{4-2}\\ =2\pm \; <!--\; plus-minus\; sign\; -->\sqrt{2}\\ x=2\pm \; <!--\; plus-minus\; sign\; -->\sqrt{2},y=2\mp \; <!--\; latex:\; \backslash mp\; -->\sqrt{2}\end{array}$
    5. 
       
       $\begin{array}{}{\left(x+y\right)}^2-xy=21\\ 25-xy=21\\ xy=4\\ t^2-5t+4=0\\ \left(t-1\right)\left(t-4\right)=0\\ x=1,y=4\\ x=4,y=1\end{array}$
    6. 
       
       $\begin{array}{}y=\frac{5-3x}{4}\\ 4x^2+\frac{5}{4}x-\frac{3}{4}{x}^2-3\frac{25+9x^2-30x}{16}=0\\ 64x^2+20x-12x^2-75-27x^2+90x=0\\ 25x^2+110x-75=0\\ 5x^2+22x-15=0\\ \left(x+5\right)\left(5x-3\right)=0\\ x=-5,y=\frac{5+15}{4}=5\\ x=\frac{3}{5},y=\frac{5-\frac{9}{5}}{4}=\frac{25-9}{4·\; <!--\; middle\; dot\; -->5}=\frac{16}{20}=\frac{4}{5}\end{array}$

コード(Emacs)

Python 3

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

print('32.')

x, y = symbols('x, y')

eqss = [(x - y - 1, x ** 2 + y ** 2 - 25),
        (y - sqrt(3) * x, x ** 2 + y ** 2 - 48),
        (y - (2 * x - 1), y ** 2 - x ** 2 - 5),
        (x + y - 4, x * y - 2),
        (x + y - 5, x ** 2 + x * y + y ** 2 - 21),
        (3 * x + 4 * y - 5, 4 * x ** 2 + x * y - 3 * y ** 2)]

for i, eqs in enumerate(eqss, 1):
    print(f'({i})')
    pprint(solve(eqs, dict=True))
    print()

p = plot(x - 1, sqrt(25 - x ** 2), -sqrt(25 - x ** 2), -4, 3,
         legend=True, show=False)
colors = ['red', 'green', 'green', 'blue', 'orange']

for i, color in enumerate(colors):
    p[i].line_color = color

p.save('sample32.png')

入出力結果(Terminal, Jupyter(IPython))

$ ./sample32.py
32.
(1)
[{x: -3, y: -4}, {x: 4, y: 3}]

(2)
[{x: -2⋅√3, y: -6}, {x: 2⋅√3, y: 6}]

(3)
[{x: -2/3, y: -7/3}, {x: 2, y: 3}]

(4)
[{x: -√2 + 2, y: √2 + 2}, {x: √2 + 2, y: -√2 + 2}]

(5)
[{x: 1, y: 4}, {x: 4, y: 1}]

(6)
[{x: -5, y: 5}, {x: 3/5, y: 4/5}]

$

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="0.005">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<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="sample32.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 => x - 1, 'red'],
           [x => Math.sqrt(25 - x ** 2), 'green'],
           [x => -Math.sqrt(25 - x ** 2), 'green']];
    
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 = [[-3, y1, -3, y2, 'blue'],
                 [x1, -4, x2, -4, 'blue'],
                 [4, y1, 4, y2, 'brown'],
                 [x1, 3, x2, 3, 'brown']];
    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 =