{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "toc": "true"
   },
   "source": [
    "# Table of Contents\n",
    " <p><div class=\"lev1 toc-item\"><a href=\"#1(a)\" data-toc-modified-id=\"1(a)-1\"><span class=\"toc-item-num\">1&nbsp;&nbsp;</span>1(a)</a></div><div class=\"lev2 toc-item\"><a href=\"#b\" data-toc-modified-id=\"b-11\"><span class=\"toc-item-num\">1.1&nbsp;&nbsp;</span>b</a></div><div class=\"lev1 toc-item\"><a href=\"#2(a)\" data-toc-modified-id=\"2(a)-2\"><span class=\"toc-item-num\">2&nbsp;&nbsp;</span>2(a)</a></div><div class=\"lev2 toc-item\"><a href=\"#2(b)\" data-toc-modified-id=\"2(b)-21\"><span class=\"toc-item-num\">2.1&nbsp;&nbsp;</span>2(b)</a></div><div class=\"lev1 toc-item\"><a href=\"#3(a)\" data-toc-modified-id=\"3(a)-3\"><span class=\"toc-item-num\">3&nbsp;&nbsp;</span>3(a)</a></div><div class=\"lev2 toc-item\"><a href=\"#3(b)\" data-toc-modified-id=\"3(b)-31\"><span class=\"toc-item-num\">3.1&nbsp;&nbsp;</span>3(b)</a></div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "t = symbols('t')\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x = cos(t)+t*sin(t)\n",
    "y = sin(t)-t*cos(t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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41Yp9zCnLRKWCm//2ETuae3nhxtlR/bQSIp6tWLGCW265hWAwyLe+9S3uvPPO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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d6f2f28>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sympy import *\n",
    "from sympy.plotting import plot_parametric\n",
    "\n",
    "p = plot_parametric(x,y,(t,-2*pi,2*pi), show=False)\n",
    "p.show()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "dx_dt = diff(x,t)\n",
    "dy_dt = diff(y,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADcAAAAxBAMAAACMrCxoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\frac{\\sin{\\left (t \\right )}}{\\cos{\\left (t \\right )}}$$"
      ],
      "text/plain": [
       "sin(t)\n",
       "──────\n",
       "cos(t)"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dy_dx = dy_dt/dx_dt\n",
    "dy_dx"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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uLkU3ehkndWIQXOQ3deHWrVv4+OOPAdDH3KsXDWzVs4emDaGhoZgwYQLKa+RQ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\nHFkIvdYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10d6f2d30>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "\n",
    "from sympy import *\n",
    "init_printing()\n",
    "x = symbols('x')\n",
    "\n",
    "f11=sin(x)*sin(x)\n",
    "f23=sin(2*x)*sin(3*x)\n",
    "f24=sin(2*x)*sin(4*x)\n",
    "\n",
    "\n",
    "p = plot(f11, f23, f24, (x, -pi, pi), ylim=[-1,1],\n",
    "         legend=True, show=False)\n",
    "p[0].line_color = 'b'\n",
    "p[1].line_color = 'r'\n",
    "p.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$n\\neq m$の場合は，位相のずれのために$0\\dots-\\pi$あるいは$-\\pi \\dots 0$において正の図形と負の図形がちょうど同じとなるため，積分はゼロとなる．一方$n\\neq m$の場合はそれぞれの領域で正となる図形しか現れない．従って，面積を有する．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 2(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡3  0   2⎤\n",
      "⎢        ⎥\n",
      "⎢0  -3  0⎥\n",
      "⎢        ⎥\n",
      "⎣2  0   3⎦\n"
     ]
    }
   ],
   "source": [
    "from sympy import *\n",
    "\n",
    "A=Matrix([[1,0,-2],[0,2,0],[-2,0,1]])\n",
    "B=Matrix([[3,0,2],[0,-3,0],[2,0,3]])\n",
    "\n",
    "pprint(B)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIkAAABLCAMAAABz2lREAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-1 & 0 & -4\\\\0 & -6 & 0\\\\-4 & 0 & -1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-1  0   -4⎤\n",
       "⎢          ⎥\n",
       "⎢0   -6  0 ⎥\n",
       "⎢          ⎥\n",
       "⎣-4  0   -1⎦"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A*B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIkAAABLCAMAAABz2lREAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-1 & 0 & -4\\\\0 & -6 & 0\\\\-4 & 0 & -1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-1  0   -4⎤\n",
       "⎢          ⎥\n",
       "⎢0   -6  0 ⎥\n",
       "⎢          ⎥\n",
       "⎣-4  0   -1⎦"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B*A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡1  0  -1⎤\n",
      "⎢        ⎥\n",
      "⎢0  1  0 ⎥\n",
      "⎢        ⎥\n",
      "⎣1  0  1 ⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAABLCAMAAABEOFk1AAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-2 & 0 & 0\\\\0 & 2 & 0\\\\0 & 0 & 6\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-2  0  0⎤\n",
       "⎢        ⎥\n",
       "⎢0   2  0⎥\n",
       "⎢        ⎥\n",
       "⎣0   0  6⎦"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PA,D = A.diagonalize()\n",
    "pprint(PA)\n",
    "PA.T*A*PA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "⎡0  -1  1⎤\n",
      "⎢        ⎥\n",
      "⎢1  0   0⎥\n",
      "⎢        ⎥\n",
      "⎣0  1   1⎦\n"
     ]
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHMAAABLCAMAAAB0tEgUAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nfKigVGOt8H09/mGPvolQSvwGDKAvCHB+TA8AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-3 & 0 & 0\\\\0 & 2 & 0\\\\0 & 0 & 10\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-3  0  0 ⎤\n",
       "⎢         ⎥\n",
       "⎢0   2  0 ⎥\n",
       "⎢         ⎥\n",
       "⎣0   0  10⎦"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PB,D = B.diagonalize()\n",
    "pprint(PB)\n",
    "PB.T*B*PB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHMAAABLCAMAAAB0tEgUAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}10 & 0 & 0\\\\0 & -3 & 0\\\\0 & 0 & 2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡10  0   0⎤\n",
       "⎢         ⎥\n",
       "⎢0   -3  0⎥\n",
       "⎢         ⎥\n",
       "⎣0   0   2⎦"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PA.T*B*PA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "PAとPBでは行列の列要素が食い違っている．しかし，行列の固有値には順番がないので，食い違いは問題なく，どちらの直交化行列PA, PBを使ってもA,Bは同時に対角化される．"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & 2 & 2\\\\2 & 2 & 0\\\\2 & 0 & 4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3  2  2⎤\n",
       "⎢       ⎥\n",
       "⎢2  2  0⎥\n",
       "⎢       ⎥\n",
       "⎣2  0  4⎦"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "\n",
    "\n",
    "A = Matrix([[3,2,2],[2,2,0],[2,0,4]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "x,y,z = symbols('x y z')\n",
    "xx = Matrix([x,y,z])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQ4AAAAaBAMAAAC0r6+vAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nrlrCG6MAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}3 x^{2} + 4 x y + 4 x z + 2 y^{2} + 4 z^{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   2                      2      2⎤\n",
       "⎣3⋅x  + 4⋅x⋅y + 4⋅x⋅z + 2⋅y  + 4⋅z ⎦"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(xx.T*A*xx)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHkAAABLCAMAAABjltjdAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRM3dIruJ72ZsPtw6XAAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAmdJREFUaAXtmm1z\ngkAMhM8DqRUQLP//v1aOtwvmsjknY9spfIE5N/uQUGXp4E5D2Lx719ZMQOdOQ1E+tvO7wO4+4vww\nkk9vg26gu0hui+J2szstYieSq/HSd8N1O1F4dO6rpIbaiWQfXPo+6bX7oPKFH9JkaieS+2a09jl/\nBqVApnYiuQ42ZmRqJ5KnYTZCG7txOyf1TO0w+Tp0T/7pBUhe7TC5yfptg+TVjpCrulm3ev4aF0W6\nQUaPyJsdITOMUgAzcnSdIztAPo/gtuUY/Jrcc2wnk0+h49KKTOxEctt0XVf48HvCt/i0KvVM7URy\nPd1H9eTC90Pt70/nMy1QO5GccLBZPsg2c9S5HNPWzclGdUzbZo46l982bRLJdS0EVU4d2zON5Hoy\nroseBFgyjeR6MqijDwIsmUZyPRnXRbdvlkwjuZ6M6xB5YmUl/Oj0pDoVeY3kkanmUKxTkddIrsFF\nGrFOQ94ieeS6O2SSvpPrduSPy+fO8vFklpnwFwNQF5G/Luz/SeJIvphq9qguIrPfKkciuYY4a2Ad\nItNIrifjOkSmkVxPRnXkQYCfth72uvIgvz67/Mpj2vkze73imPbrs8uv/OvTViZ8Ikv1HEVyPEic\n8IMHlbFkGskxGST8xYDKWPJDGt1Il8L0Hif8UEtlJmSc8AOZykzI0zSkhB/Na5HZkcWEv5FXmR1Z\nTPgbeZWZkeWEv5I3mRUZJPyFHMmMyCjhz+RYZkOGCX8iE5kJGSf8QKYynkwi+XKN0nuU8OdKKuPJ\naYjdJwfZbpbY6f9O+6feZKnGF0vKMucVCnwZJUV4k6Us3TfmdCf+AjcMTAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-2 & -1 & 2\\\\2 & -2 & 1\\\\1 & 2 & 2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-2  -1  2⎤\n",
       "⎢         ⎥\n",
       "⎢2   -2  1⎥\n",
       "⎢         ⎥\n",
       "⎣1   2   2⎦"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P,D = A.diagonalize()\n",
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 97,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "c0=P[:,0]/P[:,0].norm()\n",
    "c1=P[:,1]/P[:,1].norm()\n",
    "c2=P[:,2]/P[:,2].norm()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 102,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIEAAABNCAMAAAC2VPetAAAAVFBMVEX///8AAAAAAAAAAAAAAAAA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\nve0Jxw5ScX1pt0TlckdU17lf899AcYMPRQsAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{2}{3} & - \\frac{1}{3} & \\frac{2}{3}\\\\\\frac{2}{3} & - \\frac{2}{3} & \\frac{1}{3}\\\\\\frac{1}{3} & \\frac{2}{3} & \\frac{2}{3}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-2/3  -1/3  2/3⎤\n",
       "⎢               ⎥\n",
       "⎢2/3   -2/3  1/3⎥\n",
       "⎢               ⎥\n",
       "⎣1/3   2/3   2/3⎦"
      ]
     },
     "execution_count": 102,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "PP = c0.col_insert(1,c1).col_insert(2,c2)\n",
    "PP"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 120,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYoAAAAbBAMAAACQHdVxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{2 x}{3} - \\frac{y}{3} + \\frac{2 z}{3} & \\frac{2 x}{3} - \\frac{2 y}{3} + \\frac{z}{3} & \\frac{x}{3} + \\frac{2 y}{3} + \\frac{2 z}{3}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡  2⋅x   y   2⋅z  2⋅x   2⋅y   z  x   2⋅y   2⋅z⎤\n",
       "⎢- ─── - ─ + ───  ─── - ─── + ─  ─ + ─── + ───⎥\n",
       "⎣   3    3    3    3     3    3  3    3     3 ⎦"
      ]
     },
     "execution_count": 120,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Py = (PP*xx)\n",
    "Py.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 123,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGAAAAAaBAMAAAC6D0SsAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}3 y^{2} + 6 z^{2}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡   2      2⎤\n",
       "⎣3⋅y  + 6⋅z ⎦"
      ]
     },
     "execution_count": 123,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand(Py.T*A*Py)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 3(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 219,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.0 (Python 3.6.1-64-bit) (ground types: python)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at http://docs.sympy.org/1.0/\n"
     ]
    }
   ],
   "source": [
    "from sympy import *\n",
    "\n",
    "init_session()\n",
    "a,b,c, x = symbols('a b c x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 220,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a_b_c = 1\n",
    "ab_bc_ca = -2\n",
    "abc = -1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 221,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAAWBAMAAADa7xQeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$a^{2} + b^{2} + c^{2}$$"
      ],
      "text/plain": [
       " 2    2    2\n",
       "a  + b  + c "
      ]
     },
     "execution_count": 221,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((a+b+c)**2-2*(a*b+b*c+c*a))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 222,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzXYQMplU74mrIma7\nRN0SDTw+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAATklEQVQIHWNgVGYJY2Bg//+/gYGBsVOLgYGB\nE4jh5G5VBwYGHgOuv2DBc2AyfgHDRAaG/gsMhQwM8QYMAQwMDxkYljJwVzEwcEVMNWAAADWrDnXF\n0pOGAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$5$$"
      ],
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 222,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2_b2_c2 = a_b_c**2-2*ab_bc_ca\n",
    "a2_b2_c2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "a^2+b^2+c^2 = (a+b+c)^2 - 2(ab+bc+ca) \\\\\n",
    "= 1 - 2(-2) \\\\\n",
    "= 5 \n",
    "$$\n",
    "ア, これをa2_b2_c2とする．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 223,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAqBAMAAACw8P8rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nxd6hzIypBb65lw9Lse3+ApdsrVuuWYOMAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{a b c} \\left(a b + a c + b c\\right)$$"
      ],
      "text/plain": [
       "a⋅b + a⋅c + b⋅c\n",
       "───────────────\n",
       "     a⋅b⋅c     "
      ]
     },
     "execution_count": 223,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "together(1/a+1/b+1/c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 224,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOBAMAAAAoFKpzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAoElEQVQIHWNgVDJ2YAACVvXnQDqMgf0HiMd9\ngXUhA0MqA8NMEO8lA0M1A8MsBoZ8ByDPjIHhfgDD/gAI7xsDQ/8FkKL1AUBDPgF5B4Acju9AghFI\n+BcAGVwJIB5QDsxTAnLgKnkUQDwGoCn3gaZcYmAUAPKMGRjeBzCwKDDwgHhA27UZGKTKi7IY4jcw\ncF1gncLAsP///18gY1lrjjkwAABIUyhz5oYQCQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2.0$$"
      ],
      "text/plain": [
       "2.0"
      ]
     },
     "execution_count": 224,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1_b1_c1 = ab_bc_ca/abc\n",
    "a1_b1_c1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c} =\n",
    "\\frac{ab+bc+ca}{abc} \\\\\n",
    "= \\frac{-2}{-1}\n",
    "$$\n",
    "イ，　a1_b1_c1とする．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQsAAAAqBAMAAABW/GNGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\frac{1}{c^{2}} + \\frac{2}{b c} + \\frac{1}{b^{2}} + \\frac{2}{a c} + \\frac{2}{a b} + \\frac{1}{a^{2}}$$"
      ],
      "text/plain": [
       "1     2    1     2     2    1 \n",
       "── + ─── + ── + ─── + ─── + ──\n",
       " 2   b⋅c    2   a⋅c   a⋅b    2\n",
       "c          b                a "
      ]
     },
     "execution_count": 225,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((1/a+1/b+1/c)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 226,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAqBAMAAABfMpQVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfhHaUM4/f3QbcyVSHusXYQ1/Ajykxw2RdgiKAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{a b c} \\left(2 a + 2 b + 2 c\\right)$$"
      ],
      "text/plain": [
       "2⋅(a + b + c)\n",
       "─────────────\n",
       "    a⋅b⋅c    "
      ]
     },
     "execution_count": 226,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "together(2/(b*c)+2/(a*c)+2/(a*b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 227,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHYAAAAqBAMAAACUxbKuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\npAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{c^{2}} + \\frac{1}{b^{2}} + \\frac{1}{a^{2}}$$"
      ],
      "text/plain": [
       "1    1    1 \n",
       "── + ── + ──\n",
       " 2    2    2\n",
       "c    b    a "
      ]
     },
     "execution_count": 227,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((1/a+1/b+1/c)**2-(2/(b*c)+2/(a*c)+2/(a*b)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{1}{c^{2}} + \\frac{1}{b^{2}} + \\frac{1}{a^{2}} = \n",
    "\\left(\n",
    "\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}\n",
    "\\right)^2 - \\frac{2(a+b+c)}{abc} \\\\\n",
    "= 2^2-2*1/(-1)\n",
    "$$\n",
    "ウ，a12_b12_c12とする．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 228,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAOBAMAAADDIxFwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyEM1Embsi72ZU\n3au6f2Q3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAp0lEQVQIHT3OPQrCQBCG4TdEksWNxp/SSk/g\nFSxECEgCllrkCKm0tbP1Bm7hQXIEDyASyQmUKIgpnFVwi4GHb2ZngO5wjLzZoC91mvqJ1Yr5Hnao\nrUAbnATvZQNQBUGFMl8QFng14aK3to4zvDvxkmYuOnRwb8QV/tEqswpP6Oe/s5Wg3yL5Jahpm18m\nS3WFK3NGMtlub7oS5Y0HnInkXmdTwgUmo5IPjXwowTkCLxwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$6.0$$"
      ],
      "text/plain": [
       "6.0"
      ]
     },
     "execution_count": 228,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a12_b12_c12 = a1_b1_c1**2 - 2*(a_b_c)/(abc)\n",
    "a12_b12_c12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 229,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVkAAAAqBAMAAADrHTUzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndfhZb8lYl4c5bObIfbcp71+CImc2/AdNWAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{2} \\left(a^{2} + b^{2} + c^{2}\\right) - 6 x + \\frac{1}{c^{2}} + \\frac{1}{b^{2}} + \\frac{1}{a^{2}}$$"
      ],
      "text/plain": [
       " 2 ⎛ 2    2    2⎞         1    1    1 \n",
       "x ⋅⎝a  + b  + c ⎠ - 6⋅x + ── + ── + ──\n",
       "                           2    2    2\n",
       "                          c    b    a "
      ]
     },
     "execution_count": 229,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect(expand((a*x-1/a)**2+\n",
    "(b*x-1/b)**2+\n",
    "(c*x-1/c)**2),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIIAAAAWBAMAAAD+2lmbAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\njp/AYIbXey1KuJfJNzzs69zmP5wJdssMxwWWAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$5 x^{2} - 6 x + 6.0$$"
      ],
      "text/plain": [
       "   2            \n",
       "5⋅x  - 6⋅x + 6.0"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = a2_b2_c2 * x**2 -6*x + a12_b12_c12\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 231,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAAUBAMAAABynCdQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nbXh4xz94FyVVJlqztAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left [ -0.148331477354788, \\quad 1.34833147735479\\right ]$$"
      ],
      "text/plain": [
       "[-0.148331477354788, 1.34833147735479]"
      ]
     },
     "execution_count": 231,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(A-7,x)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "こいつはどうしようもない．Aの中に浮動小数点数が入っているので，解析的に求められない．整数の式に書き直して解く．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 232,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOQAAAA/BAMAAAACgHlSAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYbIPrm7IAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ \\frac{3}{5} + \\frac{\\sqrt{74}}{5}, \\quad - \\frac{\\sqrt{74}}{5} + \\frac{3}{5}\\right ]$$"
      ],
      "text/plain": [
       "⎡3   √74    √74   3⎤\n",
       "⎢─ + ───, - ─── + ─⎥\n",
       "⎣5    5      5    5⎦"
      ]
     },
     "execution_count": 232,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(5*x**2-6*x-6-7,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 233,
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "IPython console for SymPy 1.0 (Python 3.6.1-64-bit) (ground types: python)\n",
      "\n",
      "These commands were executed:\n",
      ">>> from __future__ import division\n",
      ">>> from sympy import *\n",
      ">>> x, y, z, t = symbols('x y z t')\n",
      ">>> k, m, n = symbols('k m n', integer=True)\n",
      ">>> f, g, h = symbols('f g h', cls=Function)\n",
      ">>> init_printing()\n",
      "\n",
      "Documentation can be found at http://docs.sympy.org/1.0/\n"
     ]
    }
   ],
   "source": [
    "from sympy import *\n",
    "\n",
    "init_session()\n",
    "a,b,c, x = symbols('a b c x')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "a_b_c = 1.1\n",
    "ab_bc_ca = -2.2\n",
    "abc = -1.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 235,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGoAAAAWBAMAAADa7xQeAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$a^{2} + b^{2} + c^{2}$$"
      ],
      "text/plain": [
       " 2    2    2\n",
       "a  + b  + c "
      ]
     },
     "execution_count": 235,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((a+b+c)**2-2*(a*b+b*c+c*a))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 236,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACIAAAAPBAMAAABzWwAZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzXYQMplU74mrIma7\nRN0SDTw+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuUlEQVQYGWNgVGYJYwABjqYWMM3A/v9/A4jF\nLMsQyMDAVszAwNipBZbiCWA4w2AZ+ZmBgRPMZ2C4vwDEYEISOQ6WAYvsVnUAcT56qF6AqOEx4PoL\nFGD+uIGhDCIC5J0Difw3YAg3AJsD5MUvABL/GBj2XwCLTGRg6AcawPAQKLIBLFIIVGMAFJkOVxMA\nlmZg8AeaAzF5KQN3FYO9AANnAzPULq6IqQYMfBMYGFojHRhYp3+cDTQBDQAAf3gyAW+eqA0AAAAA\nSUVORK5CYII=\n",
      "text/latex": [
       "$$5.61$$"
      ],
      "text/plain": [
       "5.61"
      ]
     },
     "execution_count": 236,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2_b2_c2 = a_b_c**2-2*ab_bc_ca\n",
    "a2_b2_c2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "a^2+b^2+c^2 = (a+b+c)^2 - 2(ab+bc+ca) \\\\\n",
    "= 1 - 2(-2) \\\\\n",
    "= 5 \n",
    "$$\n",
    "ア, これをa2_b2_c2とする．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 237,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAqBAMAAACw8P8rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nxd6hzIypBb65lw9Lse3+ApdsrVuuWYOMAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{a b c} \\left(a b + a c + b c\\right)$$"
      ],
      "text/plain": [
       "a⋅b + a⋅c + b⋅c\n",
       "───────────────\n",
       "     a⋅b⋅c     "
      ]
     },
     "execution_count": 237,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "together(1/a+1/b+1/c)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 238,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOBAMAAAAoFKpzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIpm7MhCriUTv3c12\nVGZoascqAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAoElEQVQIHWNgVDJ2YAACVvXnQDqMgf0HiMd9\ngXUhA0MqA8NMEO8lA0M1A8MsBoZ8ByDPjIHhfgDD/gAI7xsDQ/8FkKL1AUBDPgF5B4Acju9AghFI\n+BcAGVwJIB5QDsxTAnLgKnkUQDwGoCn3gaZcYmAUAPKMGRjeBzCwKDDwgHhA27UZGKTKi7IY4jcw\ncF1gncLAsP///18gY1lrjjkwAABIUyhz5oYQCQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$2.0$$"
      ],
      "text/plain": [
       "2.0"
      ]
     },
     "execution_count": 238,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1_b1_c1 = ab_bc_ca/abc\n",
    "a1_b1_c1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c} =\n",
    "\\frac{ab+bc+ca}{abc} \\\\\n",
    "= \\frac{-2}{-1}\n",
    "$$\n",
    "イ，　a1_b1_c1とする．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 239,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAQsAAAAqBAMAAABW/GNGAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nRK5CYII=\n",
      "text/latex": [
       "$$\\frac{1}{c^{2}} + \\frac{2}{b c} + \\frac{1}{b^{2}} + \\frac{2}{a c} + \\frac{2}{a b} + \\frac{1}{a^{2}}$$"
      ],
      "text/plain": [
       "1     2    1     2     2    1 \n",
       "── + ─── + ── + ─── + ─── + ──\n",
       " 2   b⋅c    2   a⋅c   a⋅b    2\n",
       "c          b                a "
      ]
     },
     "execution_count": 239,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((1/a+1/b+1/c)**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 240,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAqBAMAAABfMpQVAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfhHaUM4/f3QbcyVSHusXYQ1/Ajykxw2RdgiKAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\frac{1}{a b c} \\left(2 a + 2 b + 2 c\\right)$$"
      ],
      "text/plain": [
       "2⋅(a + b + c)\n",
       "─────────────\n",
       "    a⋅b⋅c    "
      ]
     },
     "execution_count": 240,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "together(2/(b*c)+2/(a*c)+2/(a*b))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 241,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHYAAAAqBAMAAACUxbKuAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\npAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\frac{1}{c^{2}} + \\frac{1}{b^{2}} + \\frac{1}{a^{2}}$$"
      ],
      "text/plain": [
       "1    1    1 \n",
       "── + ── + ──\n",
       " 2    2    2\n",
       "c    b    a "
      ]
     },
     "execution_count": 241,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expand((1/a+1/b+1/c)**2-(2/(b*c)+2/(a*c)+2/(a*b)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\frac{1}{c^{2}} + \\frac{1}{b^{2}} + \\frac{1}{a^{2}} = \n",
    "\\left(\n",
    "\\frac{1}{a}+\\frac{1}{b}+\\frac{1}{c}\n",
    "\\right)^2 - \\frac{2(a+b+c)}{abc} \\\\\n",
    "= 2^2-2*1/(-1)\n",
    "$$\n",
    "ウ，a12_b12_c12とする．"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 242,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAAAOBAMAAADDIxFwAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXYyEM1Embsi72ZU\n3au6f2Q3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAp0lEQVQIHT3OPQrCQBCG4TdEksWNxp/SSk/g\nFSxECEgCllrkCKm0tbP1Bm7hQXIEDyASyQmUKIgpnFVwi4GHb2ZngO5wjLzZoC91mvqJ1Yr5Hnao\nrUAbnATvZQNQBUGFMl8QFng14aK3to4zvDvxkmYuOnRwb8QV/tEqswpP6Oe/s5Wg3yL5Jahpm18m\nS3WFK3NGMtlub7oS5Y0HnInkXmdTwgUmo5IPjXwowTkCLxwAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$6.0$$"
      ],
      "text/plain": [
       "6.0"
      ]
     },
     "execution_count": 242,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a12_b12_c12 = a1_b1_c1**2 - 2*(a_b_c)/(abc)\n",
    "a12_b12_c12"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 243,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAVkAAAAqBAMAAADrHTUzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ndfhZb8lYl4c5bObIfbcp71+CImc2/AdNWAAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x^{2} \\left(a^{2} + b^{2} + c^{2}\\right) - 6 x + \\frac{1}{c^{2}} + \\frac{1}{b^{2}} + \\frac{1}{a^{2}}$$"
      ],
      "text/plain": [
       " 2 ⎛ 2    2    2⎞         1    1    1 \n",
       "x ⋅⎝a  + b  + c ⎠ - 6⋅x + ── + ── + ──\n",
       "                           2    2    2\n",
       "                          c    b    a "
      ]
     },
     "execution_count": 243,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "collect(expand((a*x-1/a)**2+\n",
    "(b*x-1/b)**2+\n",
    "(c*x-1/c)**2),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 244,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJwAAAAWBAMAAADHvejAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nnqUxfFN24R18yiO+SAT/AKsnmmbceuUAAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$5.61 x^{2} - 6 x + 6.0$$"
      ],
      "text/plain": [
       "      2            \n",
       "5.61⋅x  - 6⋅x + 6.0"
      ]
     },
     "execution_count": 244,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = a2_b2_c2 * x**2 -6*x + a12_b12_c12\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 245,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAYIAAAAUBAMAAABynCdQAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$\\left [ -0.146578069647643, \\quad 1.21609678622518\\right ]$$"
      ],
      "text/plain": [
       "[-0.146578069647643, 1.21609678622518]"
      ]
     },
     "execution_count": 245,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(A-7,x)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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