{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The roots of quadratic equations $a x^2 + b x + c = 0$ are $  r = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "roots are positive.\r\n"
     ]
    }
   ],
   "source": [
    "a=1; b=0; c=-3;\n",
    "d=b^2-4*a*c;\n",
    "if d>=0\n",
    "    disp('roots are positive.');\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "roots are immaginary\r\n"
     ]
    }
   ],
   "source": [
    "a=1; b=1; c=3;\n",
    "d=b^2-4*a*c;\n",
    "if d>=0\n",
    "    disp('roots are positive.');\n",
    "else\n",
    "    disp('roots are immaginary');\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "roots are immaginary\n",
      "r1 = -0.500000 + i 1.658312 \n",
      "r2 = -0.500000 - i 1.658312 \n"
     ]
    }
   ],
   "source": [
    "a=1; b=1; c=3;\n",
    "d=b^2-4*a*c;\n",
    "if d>0\n",
    "    disp('roots are positive and distinct');\n",
    "    r1 = (-b+sqrt(d))/(2*a)\n",
    "    r2 = (-b-sqrt(d))/(2*a)\n",
    "elseif d==0\n",
    "    disp('roots are real and same');\n",
    "    r = -b/(2*a);\n",
    "else\n",
    "    disp('roots are immaginary');\n",
    "    re = -b/(2*a); im = sqrt(-d)/(2*a);\n",
    "    printf (\"r1 = %f + i %f \\n\",re, im);\n",
    "    printf (\"r2 = %f - i %f \\n\",re, im);\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%plot gnuplot"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "% define a quadretic function with a, b and c parameters\n",
    "f = @(x, a, b, c) a*x.^2 + b*x + c;\n",
    "\n",
    "x= -3:0.1:3;\n",
    "y1=f(x,1,1,-1); %we get two real roots \n",
    "y2=f(x,1,-2,1); %one real root\n",
    "y3=f(x,1,1,1); %no real root\n",
    "\n",
    "plot(x,y1,'r',x,y2,'g',x,y3,'b')\n",
    "hold on\n",
    "plot(x,0*x,'k',0*ones(length(x),1),10*x,'k') %plotting axises\n",
    "hold off\n",
    "axis([-3, 3, -5,20])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"images/quadratic.png\" alt=\"Quadretic functions\" style=\"width:400px;height:300px;\">"
   ]
  }
 ],
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    {
     "text": "MetaKernel Magics",
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