Solver Quadratic Formula

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Source code of 'Quadratic Formula'

This Solver (Quadratic Formula) was created by by jim_thompson5910(13794)

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About jim_thompson5910:

==section input
Enter the coefficients of the quadratic in the form of {{{ax^2+bx+c=0}}} (note: a cannot equal zero). For instance, if you want to solve the quadratic {{{x^2+2x+3=0}}} enter {{{a=1}}}, {{{b=2}}}, and {{{c=3}}}

a=*[input a=1]  b=*[input b=2]  c=*[input c=3]

Make sure you enter whole numbers for a, b, and c. If you have something like {{{0.2x^2+5x+6=0}}} multiply both sides by 10 to get {{{2x^2+50x+60=0}}} which makes every coefficient whole. If you have something like {{{(1/2)x^2+(2/3)x+6=0}}} multiply both sides by the LCD 6 to get {{{3x^2+4x+36=0}}} which makes every coefficient whole.

and enter a variable to solve for

variable: *[input variable=x]

note: This solver is far from perfect. If you see an error in either the steps or the solution, please let me know.

Also, email me to send me any solver suggestions you would like to see on the site.

==section solution

perl

$test="$a, $b, $c";

if(($test=~m/\//)&&($test=~m/\./))
{
print "

Make sure you enter whole numbers for a, b, and c. If you have something like {{{0.2x^2+5x+6=0}}} multiply both sides by 10 to get {{{2x^2+50x+60=0}}} which makes every coefficient whole. If you have something like {{{(1/2)x^2+(2/3)x+6=0}}} multiply both sides by the LCD 6 to get {{{3x^2+4x+36=0}}} which makes every coefficient whole.

";
return;
}

if((($a=~/\./)||($b=~/\./))||($c=~/\./))
{print "

Once again make sure you enter whole numbers for a, b, and c. If you have something like {{{0.2x^2+5x+6=0}}} multiply both sides by 10 to get {{{2x^2+50x+60=0}}} which makes every coefficient whole.

";
return;}

if((($a=~/\//)||($b=~/\//))||($c=~/\//))
{print "Once again make sure you enter whole numbers for a, b, and c. If you have something like {{{(1/2)x^2+(2/3)x+6=0}}} multiply both sides by the LCD 6 to get {{{3x^2+4x+36=0}}} which makes every coefficient whole.

";
return;}

sub square_root{

my ($sq,$input,$input1,$num,$count,@gcf,$ret,$tick,$square_root_value,$num1,$num2,$sq1,$temp);

$input=$_[0];

#my ($sq);

$sq= sqrt($input);

$square_root_value=sqrt($input);

$value=$input;

$input1="sqrt($input)";

#$input="sqrt(75)";

if($input=~ m/(\d+)/)
{$num=$1;}

if($sq!~/\./)
{
$ret="$sq";
return $ret;
}

$tick=0;

for($count=1;$count<$num;$count++)
{

if($num%$count==0)
  {$gcf[$tick]=$count;
  $tick++;}
}

if($tick==0)
{$ret="$input1";
return $ret;}

for($count=1;$count<($size=@gcf);$count++)
{
 $sq=sqrt($gcf[$count]);
 if($sq!~/\./)
 {#print "\n $gcf[$count] is a perfect square \n";
 $num1=$gcf[$count]}
}

if($num1==0)
{$ret="$input1";
return $ret;
}

if($num1!=0)
{$num2=$num/$num1;

$sq1=sqrt($num2);

$sq=sqrt($num1);

if($sq1!~/\./)
{

$temp=$sq*$sq1;

$ret="$temp";

return $ret;
}

if($sq1=~/\./)
{

$ret="$sq*sqrt($num2)";

return $ret;
}

}
}

###################################

sub mult{

my ($split,@arr1,@arr2,$temp1,$temp2, @hold);

$split=$_[0];
@hold=split(/;/,$split);
@arr1=split(/,/,$hold[0]);
$arr1[2]=$arr1[1];
$arr1[1]="/";

@arr2=split(/,/,$hold[1]);
$arr2[2]=$arr2[1];
$arr2[1]="/";

#print "arr1=@arr1\n\n";

$temp1=$arr1[0]*$arr2[0];
$temp2=$arr1[2]*$arr2[2];

#$temp1=$numer1*$numer2;
#$temp2=$denom1*$denom2;
return ($temp1,"/",$temp2);
}

sub add{

my ($split,@arr1,@arr2, @hold, @tempfrac, $gcf);

$split=$_[0];
@hold=split(/;/,$split);
@arr1=split(/,/,$hold[0]);
$arr1[2]=$arr1[1];
$arr1[1]="/";

@arr2=split(/,/,$hold[1]);
$arr2[2]=$arr2[1];
$arr2[1]="/";

if(($arr1[2]!=$arr2[2])&&(($arr1[2]!=0)&&($arr2[2]!=0)))
{
 for(my $count=1;$count<abs($arr1[2]*$arr2[2]);$count++)
 {if(($arr1[2]%$count==0)&&($arr2[2]%$count==0))
 {$gcf=$count;}
 }

my $lcm=($arr1[2]*$arr2[2])/$gcf;
my $temp=$lcm/$arr1[2];

$exported_lcm=$lcm;
$exported_gcf=$gcf;

$arr1[0]=($lcm/$arr1[2])*$arr1[0];
$arr2[0]=($lcm/$arr2[2])*$arr2[0];

$arr1[2]=($lcm/$arr1[2])*$arr1[2];
$arr2[2]=($lcm/$arr2[2])*$arr2[2];
}
$tempfrac1[0]=$arr1[0]+$arr2[0];
$tempfrac2[0]=$arr1[2];
return ($tempfrac1[0],"/",$tempfrac2[0]);
}

sub subtract{

my ($split,@arr1,@arr2, @hold, @tempfrac, $gcf);

$split=$_[0];
@hold=split(/;/,$split);
@arr1=split(/,/,$hold[0]);
$arr1[2]=$arr1[1];
$arr1[1]="/";

@arr2=split(/,/,$hold[1]);
$arr2[2]=$arr2[1];
$arr2[1]="/";

if(($arr1[2]!=$arr2[2])&&(($arr1[2]!=0)&&($arr2[2]!=0)))
{
 for($count=1;$count<abs($arr1[2]*$arr2[2]);$count++)
 {if(($arr1[2]%$count==0)&&($arr2[2]%$count==0))
 {$gcf=$count;}
 }
my $lcm=($arr1[2]*$arr2[2])/$gcf;
my $temp=$lcm/$arr1[2];

$exported_lcm=$lcm;
$exported_gcf=$gcf;

$arr1[0]=($lcm/$arr1[2])*$arr1[0];
$arr2[0]=($lcm/$arr2[2])*$arr2[0];

$arr1[2]=($lcm/$arr1[2])*$arr1[2];
$arr2[2]=($lcm/$arr2[2])*$arr2[2];
}
$tempfrac1[0]=$arr1[0]-$arr2[0];
$tempfrac2[0]=$arr1[2];
return ($tempfrac1[0],"/",$tempfrac2[0]);
}

sub reduce{

my (@arr1, @original_arr1, $flag);

#$arr1[0]=$_[0];
#$arr1[1]="/";
#$arr1[2]=$_[1];
@arr1=@_;
$arr1[2]=$arr1[1];
$arr1[1]="/";

@original_arr1=@arr1;

$flag=$_[2];

if($arr1[0]==0)
{return $arr1[0];}

if($arr1[2]!=0)
{
$store=$arr1[0]/$arr1[2];

if($store>0)
{
  $arr1[0]=abs($arr1[0]);
  $arr1[2]=abs($arr1[2]);
}

if(($store<0)&&($arr1[2]<0))
{
 $arr1[0]=0-$arr1[0];
 $arr1[2]=abs($arr1[2]);
}

if($arr1[0]%$arr1[2]==0)
{
  if($flag==1)
   {$arr1[0]=$arr1[0]/$arr1[2];
   return $arr1[0];}

if($flag==2)
   {$arr1[0]=$arr1[0]/$arr1[2];
   $arr1[2]=1;
   return ($arr1[0],"/",$arr1[2]);}
#  if($flag==0)
#   {return $quotient;}

}

for (my $i=abs($arr1[0]*$arr1[2]); $i>1; $i--)
{
 if (($arr1[0]%$i==0)&&($arr1[2]%$i==0))
    {
       $arr1[0]/=$i;
       $arr1[2]/=$i;
       my $j=$i;

}
}

if($original_arr1[0]==$arr1[0])
{$exported_no_reduce=1;}

return ($arr1[0],"/",$arr1[2]);
}

sub invert{

my(@arr1,$temp);

@arr1=@_;

$temp=$arr1[0];
$arr1[0]=$arr1[1];
$arr1[1]=$temp;

return ($arr1[0], "/", $arr1[1]);
}

sub divide{

my ($split,@arr1,@arr2,$temp1,$temp2, @hold);

$split=$_[0];
@hold=split(/;/,$split);
@arr1=split(/,/,$hold[0]);
$arr1[2]=$arr1[1];
$arr1[1]="/";

@arr2=split(/,/,$hold[1]);
$arr2[2]=$arr2[1];
$arr2[1]="/";

#print "arr1=@arr1\n\n";

$temp1=$arr1[0]*$arr2[2];
$temp2=$arr1[2]*$arr2[0];

#$temp1=$numer1*$numer2;
#$temp2=$denom1*$denom2;
return ($temp1,"/",$temp2);
}

###################################

if($a==0)
{print "ERROR: NOT A QUADRATIC

When a=0, this means we have the equation

{{{0x^2+$b $variable +$c=0}}}

which becomes

{{{$b $variable+$c=0}}}

which is a linear equation and not a quadratic. So that means the quadratic formula does not apply.

";
return undef;}

#################### ERROR AND INPUT CHECK
my @temp=($a,$b,$c);
@temp=map{s/\-//g;$_} @temp;

for(my $i=0; $i<@temp; $i++)
{
 if($temp[$i]=~m/\D/)
 {
 print "Error: Enter WHOLE numbers only.";
 print "
 If you need to enter fractions, multiply EVERY term by the LCD to make the numbers whole. If you need to enter decimal values, multiply EVERY number by some factor of 10 (to move the decimal point) to make the values whole.
 ";
 return;
 }
}

###############Convert decimal to whole (insert back in if needed)

## EXTRA STUFF
$extra_toggle=0;
$vertex=-$b/(2*$a);
$negated_eval=$a*(-$b/(2*$a))**2+$b*(-$b/(2*$a))+$c;
$negated_original="-$a*(x-$vertex)^2+$negated_eval";
$negated_original_string=$negated_original;
$negated_original_string=~s/\-\-/\+/g;
$negated_original_string=~s/\+\-/\-/g;

if((-$negated_eval/-$a)>0)
{$negated_root1=sqrt(-$negated_eval/-$a)+$vertex;
$negated_root2=-sqrt(-$negated_eval/-$a)+$vertex;}
#$negated_root2=(-$negated_b-sqrt($negated_b**2-4*$negated_a*$negated_c))/(2*$negated_a);

$complex_radius=abs($negated_root1-$negated_root2)/2;
$complex_center=($negated_root1+$negated_root2)/2;

$original_string="$a*$variable^2+$b*$variable+$c";
$original_string_with_zero=$original_string;
#$original_string=~s/\D0\*x//;  ## needs work?
$original_string=~s/\D0\*[a-z]//;  ## needs work?
$original_string=~s/\+\+/\+/g;
$original_string=~s/\+\-/\-/g;
$original_string=~s/^1\*//g;
$original_string=~s/\-1\*/\-/g;
$original_string=~s/\+1\*/\+/g;

$original_graph=$original_string;
$original_graph=~s/[a-z]/x/g;

$original_string_with_zero=~s/\+\+/\+/g;
$original_string_with_zero=~s/\+\-/\-/g;
$original_string_with_zero=~s/^1\*//g;
$original_string_with_zero=~s/\-1\*/\-/g;
$original_string_with_zero=~s/\+1\*/\+/g;

print "Let's use the quadratic formula to solve for $variable:\n\n";

$start_string_quadratic=

print "

Starting with the general quadratic

{{{a$variable^2+b$variable+c=0}}}

the general solution using the quadratic equation is:

{{{$variable = (-b +- sqrt( b^2-4*a*c ))/(2*a)}}}";

$equa="-$b +- sqrt( ($b)^2-4*$a*$c ))/(2*$a)";

$equa=~s/^\-/q/;
$equa=~s/\-\-/\+/g;
$equa=~s/^\+//;
$equa=~s/q/\-/;

$equa=~s/^\-0/0/; ## remove -0

if(($b==0)||($c==0))
{
 if($b==0)
  {
   $note1="(note: since the polynomial does not have an \"x\" term, the 2nd coefficient is zero. In other words, b=0. So that means the polynomial really looks like {{{$original_string_with_zero=0}}} ";}

if($c==0)
  {
   $note1="(note: since the polynomial does not have a constant term, the 3rd coefficient is zero. In other words, c=0. So that means the polynomial really looks like {{{$original_string_with_zero=0}}} ";}

if(($b==0)&&($c==0))
  {
   $note1="(note: since the polynomial does not have an \"x\" term or a constant term, the 2nd and 3rd coefficient are zero. In other words, b=0 and c=0. So that means the polynomial really looks like {{{$original_string_with_zero=0}}} ";}

}
else
 {$note1="(";}

print "

So lets solve {{{$original_string=0}}} $note1 notice {{{a=$a}}}, {{{b=$b}}}, and {{{c=$c}}})

{{{$variable = ($equa}}} Plug in a=$a, b=$b, and c=$c

if($equa=~m/\-\-/)
{$equa=~s/\-\-/\+/g;
$equa=~s/^\+//;

#$test=abs($b);
print "

{{{$variable = ($equa}}} Negate $b to get ", abs($b),"

";}

$b_temp=$b**2;
$c_temp=-4*$c*$a;

$equa="-$b +- sqrt( $b_temp-4*$a*$c ))/(2*$a)";

$equa=~s/\-\-/\+/g;
$equa=~s/^\+//;

$equa=~s/^\-0/0/; ## remove -0

print "

{{{$variable = ($equa}}} Square $b to get $b_temp ", ($b<0)? "(note: remember when you square $b, you must square the negative as well. This is because {{{($b)^2=$b*$b=$b_temp}}}.)":"";

print"

$equa="-$b +- sqrt( $b_temp+$c_temp ))/(2*$a)";

$equa=~s/\-\-/\+/g;
$equa=~s/^\+//;

$equa=~s/^\-0/0/; ## remove -0

print "

{{{$variable = ($equa}}} Multiply {{{-4*$c*$a}}} to get {{{$c_temp}}}

$add=(-4*$c*$a)+$b_temp;

$equa="-$b +- sqrt( $add ))/(2*$a)";

$equa=~s/\-\-/\+/g;
$equa=~s/^\+//;

$equa=~s/^\-0/0/; ## remove -0

print "

{{{$variable = ($equa}}} Combine like terms in the radicand (everything under the square root)

if($add<0)
{

#$imaginary="i";
 $original_add=$add;
 $add=abs($add);
 $radicand1=square_root($add);
 $radicand1=~s/sqrt/i*sqrt/;

if($radicand1!~/sqrt/)
 {$radicand1="$radicand1"."*i";}

$equa="-$b +- $radicand1)/(2*$a)";

$equa=~s/\-\-/\+/g;
$equa=~s/^\+//;

$equa=~s/^\-0/0/; ## remove -0

# since we cannot take the square root of a negative value, we must factor {{{sqrt($original_add)}}} #to {{{i*sqrt($add)}}} to make the radicand positive.

print "

{{{$variable = ($equa}}} Simplify the square root (note: If you need help with simplifying the square root, check out this <a href=http://www.algebra.com/algebra/homework/Radicals/simplifying-square-roots.solver> solver</a>)

$equa="-$b +- $radicand1)/(".2*$a.")";

$equa=~s/\-\-/\+/g;
$equa=~s/^\+//;

$equa=~s/^\-0/0/; ## remove -0

print "

{{{$variable = ($equa}}} Multiply 2 and $a to get ", 2*$a ,"

if($radicand1=~/sqrt/)
 {
  if($radicand1=~m/(\d+)\*i\*sqrt\((\d+)/)
  {#@arr1=($1,"/",2*$a);
  @temp=reduce($1,2*$a);
  $square_root_value="$temp[0]*sqrt($2)/$temp[2]";
  $square_root_value=~s/1\*//;}

if(($radicand1!~m/(\d+)\*i\*sqrt\((\d+)/)&&($radicand1=~m/i\*sqrt\((\d+)/))
  {#@arr1=(1,"/",2*$a);
  @temp=reduce(1,2*$a,1);
  $square_root_value="$temp[0]*sqrt($1)/$temp[2]";
  $square_root_value=~s/1\*//;}
 }
else
{
  if($radicand1=~m/(\d+)\*i/)
  {#@arr1=($1,"/",2*$a);
  @temp=reduce($1,2*$a,1);
  $square_root_value="@temp";
  #$square_root_value="$arr1[0]/$arr1[2]";
  $square_root_value=~s/1\*//;}
}

#@arr1=(-$b,"/",2*$a);
@temp=reduce(-$b,2*$a,1);
$first="@temp";
$first=~s/\s//g;

$root1="$first + $square_root_value*i";
$root2="$first - $square_root_value*i";

$root1=~s/\+\-/\-/g;
$root1=~s/\-\-/\+/g;

$root2=~s/\+\-/\-/g;
$root2=~s/\-\-/\+/g;

$root1=~s/(\D)1\*/\1/g;
$root2=~s/(\D)1\*/\1/g;

$x_vertex=-$b/(2*$a);
$y_vertex=$a*$x_vertex**2+$b*$x_vertex+$c;

$root1=~s/\/i/i/;
$root2=~s/\/i/i/;

$root1=~s/\s//g;
$root2=~s/\s//g;

$root1=~s/\+\-/\-/g;
$root1=~s/\-\-/\+/g;
$root2=~s/\+\-/\-/g;
$root2=~s/\-\-/\+/g;

$x_min=$x_vertex-15;
$x_max=$x_vertex+15;

$y_min=$y_vertex-15;
$y_max=$y_vertex+15;

print "

After simplifying, the quadratic has roots of

{{{$variable=$root1}}} or {{{$variable=$root2}}}

Notice if we graph the quadratic {{{y=$original_graph}}}";

if($variable!~m/x/)
{print " (just replace $variable with x)";}

print ", we get

{{{ graph( 500, 500, $x_min, $x_max, $y_min, $y_max, $original_graph) }}} graph of {{{y=$original_graph}}}

And we can see that there are no real roots

To visually verify the answer, check out <a href=http://www.math.hmc.edu/funfacts/ffiles/10005.1.shtml>this page</a> to see a visual representation of imaginary roots

#we can see that the quadratic has no x-intercepts. This means there are no real solutions. So this verifies our answer.

if($extra_toggle==1)
{
print "

EXTRA:
Reflect {{{y=$original_string}}} (red) over the dotted line {{{y=$negated_eval}}} to get {{{y=$negated_original_string}}} (green)

Draw a circle in which fits right between the roots of the reflected graph. Plot two points on the circle where the real roots are

{{{drawing( 900, 900, $x_min, $x_max, $y_min, $y_max,
 graph( 900, 900, $x_min, $x_max, $y_min, $y_max, $original_string,$negated_original),
 graph( 900, 900, $x_min, $x_max, $y_min, $y_max, $negated_eval*sqrt(sin(5x))/sqrt(sin(5x))),
 circle($complex_center,0,$complex_radius),

circle($negated_root1,0,0.08),
 circle($negated_root1,0,0.10),

circle($negated_root2,0,0.08),
 circle($negated_root2,0,0.10)
 ) }}}

Rotate this circle 90 degrees so the points are now on the top and bottom of the circle

circle($complex_center,-$complex_radius,0.08),
 circle($complex_center,-$complex_radius,0.10)
 ) }}}

Now if you use the x-axis as the real axis and the y-axis as the imaginary axis, you can see that the real component is $complex_center and the imaginary component is $complex_radius

So this means that the complex roots are \*\[Tex \\LARGE x=$complex_center \\pm $complex_radius i\]. So this visually verifies our answer.

";
}

#$negated_root1=($b+sqrt((-$b)**2-4*-$a*-$c))/(2*-$a);
#$negated_root2=($b-sqrt((-$b)**2-4*-$a*-$c))/(2*-$a);
#$complex_radius=abs($negated_root1-$negated_root2)/2;
#$complex_center=($negated_root1+$negated_root2)/2;

return undef;

}

####

$radicand=square_root($add);

$equa="-$b +- $radicand)/(2*$a)";

$equa=~s/\-\-/\+/g;
$equa=~s/^\+//;

$equa=~s/^\-0/0/; ## remove -0

print "

$a_temp=2*$a;
$equa="-$b +- $radicand)/$a_temp";
$equa1="-$b + $radicand)/$a_temp";
$equa2="-$b - $radicand)/$a_temp";
$equa=~s/\-\-/\+/g;
$equa=~s/^\+//;

$equa1=~s/\-\-/\+/g;
$equa1=~s/^\+//;

$equa2=~s/\-\-/\+/g;
$equa2=~s/^\+//;

$equa=~s/^\-0/0/; ## remove -0
$equa1=~s/^\-0/0/; ## remove -0
$equa2=~s/^\-0/0/; ## remove -0

print "

{{{$variable = ($equa}}} Multiply 2 and $a to get $a_temp

So now the expression breaks down into two parts

{{{$variable = ($equa1}}} or {{{$variable = ($equa2}}}

if($radicand!~m/sqrt/)
{$add=-$b+$radicand;
print "
Lets look at the first part:

{{{x=($equa1}}}

{{{$variable=$add/$a_temp}}} Add the terms in the numerator";
#@arr1=($add,"/",$a_temp);
@final_array=reduce($add,$a_temp);
$final="@final_array";
$final=~s/\s//g;
$final=~s/\/1$//g;

print "

{{{$variable=$final}}} Divide

So one answer is

{{{$variable=$final}}}

";
$ans[0]=$final;

$add=-$b-$radicand;
print "

Now lets look at the second part:

{{{x=($equa2}}}

{{{$variable=$add/$a_temp}}} Subtract the terms in the numerator";
#@arr1=($add,"/",$a_temp);
@final_array=reduce($add,$a_temp);
$final="@final_array";
$final=~s/\s//g;
$final=~s/\/1$//g;

print "

{{{$variable=$final}}} Divide

So another answer is

{{{$variable=$final}}}";
$ans[1]=$final;

if($ans[0]>$ans[1])
{$x_min=$ans[1]-10;
$x_max=$ans[0]+10;}

if($ans[1]>=$ans[0])
{$x_min=$ans[0]-10;
$x_max=$ans[1]+10;}

if($ans[0]!=$ans[1])
{print "

So our solutions are:

{{{$variable=$ans[0]}}} or {{{$variable=$ans[1]}}}

Notice when we graph {{{$original_graph}}}";

if($variable!~m/x/)
{print " (just replace $variable with x)";}

print ", we get:

{{{ graph( 500, 500, $x_min, $x_max, $x_min, $x_max,$a*x^2+$b*x+$c) }}}

and we can see that the roots are {{{$variable=$ans[0]}}} and {{{$variable=$ans[1]}}}. This verifies our answer";}

if($ans[0]==$ans[1])
{print "

So our solutions are:

{{{$variable=$ans[0]}}} or {{{$variable=$ans[1]}}}

which means we only have one solution

{{{x=$ans[0]}}}

Notice when we graph {{{$original_graph}}}";

if($variable!~m/x/)
{print " (just replace $variable with x)";}

print ", we get:

{{{ graph( 500, 500, $x_min, $x_max, $x_min, $x_max,$a*x^2+$b*x+$c) }}}

and we can see that the only root is {{{$variable=$ans[0]}}}. This verifies our answer";}

}

if($radicand=~m/sqrt/)
{$approx_answer[0]=(-$b + sqrt( $b**2-4*$a*$c ))/(2*$a);
$approx_answer[1]=(-$b - sqrt( $b**2-4*$a*$c ))/(2*$a);

if($approx_answer[0]>$approx_answer[1])
{$x_min=$approx_answer[1]-10;
$x_max=$approx_answer[0]+10;}

if($approx_answer[1]>=$approx_answer[0])
{$x_min=$approx_answer[0]-10;
$x_max=$approx_answer[1]+10;}

@arr1=();
#@arr1=(-$b,"/",2*$a);
@tempor1=reduce(-$b,2*$a,1);
$frac1="@tempor1";

if($radicand=~m/(\d+)\*sqrt\((\d+)/)
  {
  @arr1=();
  #@arr1=($1,"/",2*$a);
  @tempor2=reduce($1,2*$a);
  $frac2="@tempor2";
  $argument=$2;
  }
else
  {
  @arr1=();
  #@arr1=(1,"/",2*$a);
  @tempor2=reduce(1,2*$a);
  $frac2="@tempor2";
  $frac2="1/".2*$a."";}

if($radicand=~m/sqrt\((\d+)/)
{$argument=$1;}

$split1="{{{$variable=-$b/".2*$a."+$radicand/".2*$a."}}}";
$split2="{{{$variable=-$b/".2*$a."-$radicand/".2*$a."}}}";

$simplified_ans1="{{{$variable=$frac1+$tempor2[0]*sqrt($argument)/$tempor2[2]}}}";
$simplified_ans2="{{{$variable=$frac1-$tempor2[0]*sqrt($argument)/$tempor2[2]}}}";

#$simplified_ans2="{{{$variable=$frac1-$frac2*sqrt($argument)}}}";

$simplified_ans1=~s/\+\-/\-/;
$simplified_ans1=~s/\-\-/\+/;
$simplified_ans2=~s/\+\-/\-/;
$simplified_ans2=~s/\-\-/\+/;
$simplified_ans1=~s/1\*//;
$simplified_ans2=~s/1\*//;

$simplified_ans1=~s/\/1}}}$/}}}/;
$simplified_ans2=~s/\/1}}}$/}}}/;

$split1=~s/\+\-/\-/;
$split1=~s/\-\-/\+/;
$split2=~s/\+\-/\-/;
$split2=~s/\-\-/\+/;

$split1=~s/=\-0/=0/; ## remove -0 
$split2=~s/=\-0/=0/; ## remove -0

print "

Now break up the fraction

$split1 or $split2

Simplify

$simplified_ans1 or $simplified_ans2

So these expressions approximate to

{{{$variable=$approx_answer[0]}}} or {{{$variable=$approx_answer[1]}}}

So our solutions are:

{{{$variable=$approx_answer[0]}}} or {{{$variable=$approx_answer[1]}}}

Notice when we graph {{{$original_graph}}}";

if($variable!~m/x/)
{print " (just replace $variable with x)";}

print ", we get:

{{{ graph( 500, 500, $x_min, $x_max, $x_min, $x_max,$a*x^2+$b*x+$c) }}}

when we use the root finder feature on a calculator, we find that {{{x=$approx_answer[0]}}} and {{{x=$approx_answer[1]}}}.So this verifies our answer";
}

==section output

==section check
  angle=40 angle1=50