Solver Factoring Quadratics with a leading coefficient of 1 (a=1)

Source code of 'Factoring Quadratics with a leading coefficient of 1 (a=1)'

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==section input
Enter the coefficients of the quadratic in the form {{{ax^2+bx+c}}} a=*[input a=1] b=*[input b=5] c=*[input c=6] note: a must equal 1. If you have a quadratic where a>1 then take a look at this <a href="change-this-name4450.solver"> solver</a>

==section solution
perl

my $count=0, @f, $max=0, $half=0, $size=0, $ab=0, $sum, @factor, $check=0, $flag=0, $flag2, $temp1, $temp2, $string;

$ab=abs($c);

for ($count=1; $count<=$ab; $count++)
{$mod=$ab%$count;
  if($mod == 0)
   { $f[$half]=$count;
     $half++;
     }
}

$size=@f;

print "In order to factor {{{$a*x^2+$b*x+$c}}}, first we need to ask ourselves: What two numbers multiply to $c and add to $b? Lets find out by listing all of the possible factors of $c

Factors:

for ($count=0; $count<$size; $count++)
{
 print "$f[$count],";
}

print "

";
for ($count=0; $count<$size; $count++)
{
 print "-$f[$count],";
}
print "List the negative factors as well. This will allow us to find all possible combinations";

print "

print "These factors pair up to multiply to $c.

if($c>0)
{for($count=0;$count<$size/2;$count++)
{
 print "$f[$count]*$f[$size-$count-1]=$c

";
}
}

if($c>0)
{
 for($count=0;$count<$size/2;$count++)
{
 print "(-$f[$count])*(-$f[$size-$count-1])=$c

";
}
print "note: remember two negative numbers multiplied together make a positive number

";
}

if($c<0)
{
 for($count=0;$count<$size/2;$count++)
{
 print "(-$f[$count])*($f[$size-$count-1])=$c

";
}
}

print " Now which of these pairs add to $b? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to $b

print "<TABLE>";

print "<TR><TD>First Number</TD><TD>|</TD><TD>Second Number</TD><TD>|</TD><TD>Sum</TD></TR>";

if($c>0)
{
for($count=0;$count<$size/2;$count++)
{
  $sum=$f[$count]+$f[$size-$count-1];
  print "<TR><TD>$f[$count]</TD><TD>|</TD><TD>$f[$size-$count-1]</TD>|</TD><TD>|</TD><TD>$f[$count]+$f[$size-$count-1]=$sum</TD></TR>";

if($sum==$b)
     {$factor[$check]=$f[$count];
       $check++;
      $factor[$check]=$f[$size-$count-1];
      }
}
}

if($c>0)
{
for($count=0;$count<$size/2;$count++)
{
  $sum=0-($f[$count]+$f[$size-$count-1]);
  print "<TR><TD>-$f[$count]</TD><TD>|</TD><TD>-$f[$size-$count-1]</TD>|</TD><TD>|</TD><TD>-$f[$count]+(-$f[$size-$count-1])=$sum</TD></TR>";

if($sum==$b)
     {$factor[$check]=$f[$count];
       $check++;
      $factor[$check]=$f[$size-$count-1];
      }
}
}

if($c<0)
{
for($count=0;$count<$size/2;$count++)
{
  $sum=$f[$count]-$f[$size-$count-1];
  print "<TR><TD>$f[$count]</TD><TD>|</TD><TD>-$f[$size-$count-1]</TD>|</TD><TD>|</TD><TD>$f[$count]+(-$f[$size-$count-1])=$sum</TD></TR>";

if($sum==$b)
     {$factor[$check]=$f[$count];
       $flag=2;
       $check++;
      $factor[$check]=$f[$size-$count-1];
      }
}
}

if($c<0)
{
for($count=0;$count<$size/2;$count++)
{
  $sum=$f[$size-$count-1]-$f[$count];
  print "<TR><TD>-$f[$count]</TD><TD>|</TD><TD>$f[$size-$count-1]</TD>|</TD><TD>|</TD><TD>(-$f[$count])+$f[$size-$count-1]=$sum</TD></TR>";

if($sum==$b)
     {$factor[$check]=$f[$count];
       $flag=1;
       $check++;
      $factor[$check]=$f[$size-$count-1];
      }
}
}
print "<TABLE>";

if((($check>0)&&($c>0))&&($b>0))
{print "We can see from the table that $factor[0] and $factor[1] add to $b. So the two numbers that multiply to $c and add to $b are: $factor[0] and $factor[1]";
   $flag2=1;}

if((($check>0)&&($c>0))&&($b<0))
{print "We can see from the table that -$factor[0] and -$factor[1] add to $b.So the two numbers that multiply to $c and add to $b are: -$factor[0] and -$factor[1]";
 $flag2=2;}

if($check>0)
{
 if(($c<0)&&($flag==1))
   {print "We can see from the table that -$factor[0] and $factor[1] add to $b.So the two numbers that multiply to $c and add to $b are: -$factor[0] and $factor[1]";
 $flag2=3;}
 if(($c<0)&&($flag==2))
   {print "We can see from the table that $factor[0] and -$factor[1] add to $b.So the two numbers that multiply to $c and add to $b are: $factor[0] and -$factor[1]";
$flag2=4;}
}

if($check>0)
{print "

Now we substitute these numbers into a and b of the general equation of a product of linear factors which is:

{{{(x+a)(x+b)}}}"; }

if($flag2==1)
{print "substitute a=$factor[0] and b=$factor[1]

";
$string="(x+$factor[0])(x+$factor[1])";
print "So the equation becomes:

(x+$factor[0])(x+$factor[1])

Notice that if we foil (x+$factor[0])(x+$factor[1]) we get the quadratic {{{$a*x^2+$b*x+$c}}} again";}

if($flag2==2)
{print "substitute a=-$factor[0] and b=-$factor[1]

";
$string="(x-$factor[0])(x-$factor[1])";

print "So the equation becomes:

(x-$factor[0])(x-$factor[1])

Notice that if we foil (x-$factor[0])(x-$factor[1]) we get the quadratic {{{$a*x^2+$b*x+$c}}} again";}

if($flag2==3)
{print "substitute a=-$factor[0] and b=$factor[1]

";
$string="(x-$factor[0])(x+$factor[1])";
print "So the equation becomes:

(x-$factor[0])(x+$factor[1])

Notice that if we foil (x-$factor[0])(x+$factor[1]) we get the quadratic {{{$a*x^2+$b*x+$c}}} again";}

if($flag2==4)
{print "substitute a=$factor[0] and b=-$factor[1]

";
$string="(x+$factor[0])(x-$factor[1])";
print "So the equation becomes:

(x+$factor[0])(x-$factor[1])

Notice that if we foil (x+$factor[0])(x-$factor[1]) we get the quadratic {{{$a*x^2+$b*x+$c}}} again";}

if($check==0)
{
print "

None of these factors add to $b. So this quadratic cannot be factored. In order to solve for x, we need to use the quadratic formula.";

}
$temp1=$factor[0];
$temp2=$factor[1];

==section output
string

==section check
  b=5  c=6 temp1=2 temp2=3
  b=3  c=2 temp1=1 temp2=2