Formula for factorization of a Quadratic Equation?

Question

To be clear I am looking for an equation to go from

$$Ax^2 + Bx + C = 0$$

To

$$(Dx + E)(Fx + G) = 0$$

And I need it to be able to be done in a computer as it will be going in my app.

Thanks in Advance!

Answer 1

Using the naive quadratic formula has poor numeric stability properties (although it always works fine on problems you are given in 9th grade algebra). When either $A$ or $C$ is is small, then one of the roots will involve subtraction of two nearly-equal quantities, with a loss of accuracy.

The correct way to compute the roots is to first compute $$M = -\frac12\left( B+ \mbox{sign of }(B) \sqrt{B^2-4AC} \right)$$

and then the equation factors into $$A \left(x-\frac{M}{A}\right)\left(x-\frac{C}{M}\right) = 0 $$ See Numeric Recipes.

Answer 2

A straightforward way is to divide by $A$ and solve that equation with the quadratic formula. Then if your roots of that equation are $r,s$ then the factorization is

$$(x-r)(x-s)=0.$$

If you're only interested in working with equations whose roots are real and rational, then your equation will have $r=r_1/r_2$ and $s = s_1/s_2$ where $r_1, r_2, s_1, s_2$ are integers, $r_2, s_2 \neq 0$.

Then in this special case just multiply by $r_2, s_2$:

$$(r_2 x - r_1)(s_2 x - s_1) = 0.$$

Answer 3

You can just use the quadratic formula: $$ \left( x - \left(\frac{-B + \sqrt{B^2 - 4AC}}{2A}\right)\right)\left( x - \left(\frac{-B - \sqrt{B^2 - 4AC}}{2A}\right)\right)=0 $$

Answer 4

jagy@phobeusjunior:~$ ./factor 30 97 55
    ( 2 x + 5 )  ( 15 x + 11 ) 
    jagy@phobeusjunior:~$ ./factor 1 2 1
( 1 x + 1 )  ( 1 x + 1 ) 
jagy@phobeusjunior:~$ ./factor  4096 100000 390625
( 32 x + 625 )  ( 128 x + 625 ) 
jagy@phobeusjunior:~$ ./factor  4096 100000 390623
non-square discriminant  
jagy@phobeusjunior:~$ ./factor  4096 100000 3906237777
negative discriminant  
jagy@phobeusjunior:~$

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SUMMARY: with $$ \color{blue}{b^2 - 4ac = \delta^2,} $$ take $$ \color{blue}{ a_1 = \gcd \left( a, \frac{(b + \delta)}{2} \right) ; \; \; \; \; \; a_2 = \frac{a}{a_1},} $$ then $$ \color{blue}{ a x^2 + b x y + c y^2 = \; \left(a_2x+ \left( \frac{b + \delta}{2a_1} \right) y \right) \; \; \left(a_1x+ \left( \frac{b - \delta}{2a_2} \right) y \right) \; } $$ in integers. If you do not want the variable $y,$ set $$ y=1. $$

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#include <iostream>
#include <stdlib.h>
#include <fstream>
#include <strstream>
#include <list>
#include <set>
#include <math.h>
#include <iomanip>
#include <string>
#include <algorithm>
#include <iterator>
#include <gmp.h>
#include <gmpxx.h>


using namespace std;

//   g++  -o factor factor.cc  -lgmp -lgmpxx



mpz_class mp_Sqrt( mpz_class  w)
{
  if (w <= 0) return 0;
  else if (w < 2) return 1;
  else
  {
    mpz_class squirt;
    mpz_sqrt( squirt.get_mpz_t(), w.get_mpz_t()  );
    return  squirt ;
  }  // else w >= 2;
}


mpz_class mp_GCD( mpz_class  i,  mpz_class  j )
{
     if ( i < 0 ) i *= -1 ; // should work within scope
   if ( j < 0 ) j *= -1 ;

   if ( i == 0) return j;    //  gives gcd(0,0) = 0, practical choice
   if ( j == 0) return i;
   if( i < j )  return mp_GCD( i , j % i ) ;
   else  return mp_GCD( j , i % j ) ;
}





  int main(int argc, char *argv[])
{
  if ( argc != 4) cout << "Usage: ./factor A B C   " << endl;
  else {



  int aa,bb,cc;

    aa = atoi(argv[1]);
        bb = atoi(argv[2]);
    cc = atoi(argv[3]);


mpz_class a,b,c;

a = aa; b = bb; c = cc;

mpz_class discr = b * b - 4 * a * c;

if ( discr < 0 ) cout << " negative discriminant  " << endl;
else{


     mpz_class d = mp_Sqrt(discr);
  if( d * d != discr) cout << " non-square discriminant  " << endl;
  else {  
    mpz_class a1, a2;
    a1 =  mp_GCD( a, (b + d)/2 );
    a2 = a / a1;
    cout << "( " << a2 << " x + " <<   (b + d)/(2 * a1) << " )  ( " << a1 << " x + " <<   (b - d)/(2 * a2) << " ) " << endl;
  } // else square
  } // else disc >= 0
  }  // else argc

    return 0 ;
}

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Answer 5

Quantity under radical sign should be a perfect square.

Standard quadratic equation factorization is :

$$ \left( 2Ax +B + \sqrt{B^2 - 4AC}\right) \left( 2Ax +B - \sqrt{B^2 - 4AC}\right) =0 $$

The first part is the derivative and the second part, discriminant root.

Also, will this example help?

$$ m^2x^2+(m^2+n^2)x+n^2=0 $$

factors to

$$ (x+1) (m^2 x +n^2) = 0. $$