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   "source": [
    "# Solving a Quadratic Equation\n",
    "\n",
    "For\n",
    "\n",
    "$$\n",
    "ax^2+bx+c=0,\\qquad a\\ne0,\n",
    "$$\n",
    "\n",
    "the two roots are\n",
    "\n",
    "$$\n",
    "x=\\frac{-b\\pm\\sqrt{b^2-4ac}}{2a}.\n",
    "$$\n",
    "\n",
    "Rather than entering roots separately for each example, the Egison\n",
    "program reads the coefficients of a polynomial, constructs its\n",
    "discriminant, and applies one typed symbolic solver.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0c1634d",
   "metadata": {},
   "source": [
    "## Coefficients and the discriminant\n",
    "\n",
    "Egison's coefficients function returns coefficients in ascending\n",
    "degree order. Thus the pattern $[a_0,a_1,a_2]$ recognizes\n",
    "$a_2x^2+a_1x+a_0$. The helper keeps the cleared-denominator\n",
    "discriminant $b^2-4ac$ intact and explicitly groups the denominator\n",
    "as $2a$.\n"
   ]
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  {
   "cell_type": "code",
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   "source": [
    "declare symbol x, a, b, c: MathValue\n",
    "\n",
    "def solveQuadraticCoefficientsDemo\n",
    "  (a : MathValue)\n",
    "  (b : MathValue)\n",
    "  (c : MathValue)\n",
    "  : (MathValue, MathValue) :=\n",
    "  let discriminant := b ^ 2 - 4 * a * c\n",
    "   in ( ((- b) + sqrt discriminant) / (2 * a)\n",
    "      , ((- b) - sqrt discriminant) / (2 * a) )\n",
    "\n",
    "def solveQuadraticDemo\n",
    "  (f : MathValue)\n",
    "  (x : MathValue)\n",
    "  : (MathValue, MathValue) :=\n",
    "  match coefficients f x as list mathValue with\n",
    "    | [$a_0, $a_1, $a_2] ->\n",
    "      solveQuadraticCoefficientsDemo a_2 a_1 a_0\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "61c83725",
   "metadata": {},
   "source": [
    "## A cyclotomic example\n",
    "\n",
    "The polynomial $x^2+x+1$ has discriminant $-3$. Its roots are the\n",
    "two primitive cube roots of unity.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "a97642ab",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$(\\frac{1}{2} \\sqrt{3} i + \\frac{-1}{2}, \\frac{-1}{2} \\sqrt{3} i + \\frac{-1}{2})$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "solveQuadraticDemo (x ^ 2 + x + 1) x\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c9217626",
   "metadata": {},
   "source": [
    "## The symbolic formula\n",
    "\n",
    "Leaving $a$, $b$, and $c$ symbolic exposes the usual discriminant\n",
    "without any special formatting code.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "c72484c5",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$(\\frac{-1}{2} a^{-1} b + \\frac{1}{2} \\sqrt{b^{2} - 4 a c} a^{-1}, \\frac{-1}{2} a^{-1} b + \\frac{-1}{2} \\sqrt{b^{2} - 4 a c} a^{-1})$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "solveQuadraticDemo (a * x ^ 2 + b * x + c) x\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ca393c5",
   "metadata": {},
   "source": [
    "## A useful rescaling\n",
    "\n",
    "Writing the middle coefficient as $2b$ gives the equivalent compact\n",
    "form\n",
    "\n",
    "$$\n",
    "x=\\frac{-b\\pm\\sqrt{b^2-ac}}{a}.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "e43288c2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "$(-a^{-1} b + \\sqrt{b^{2} - a c} a^{-1}, -a^{-1} b - \\sqrt{b^{2} - a c} a^{-1})$"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "solveQuadraticDemo (a * x ^ 2 + 2 * b * x + c) x\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d6bfd467",
   "metadata": {},
   "source": [
    "## Takeaway\n",
    "\n",
    "Pattern matching and the typed helper separate the calculation into\n",
    "two transparent stages: read the polynomial coefficients, then form\n",
    "the discriminant and both signs of its square root. The displayed\n",
    "answers are produced from each input polynomial rather than inserted\n",
    "as precomputed outputs.\n"
   ]
  }
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