Solver Completing the Square to Get a Quadratic into Vertex Form

Source code of 'Completing the Square to Get a Quadratic into Vertex Form'

This Solver (Completing the Square to Get a Quadratic into Vertex Form) was created by by jim_thompson5910(35256)

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==section input
Enter the quadratic in the form of {{{ax^2+bx+c}}}

a=*[input my_a=1] b=*[input my_b=5] c=*[input my_c=6]. If you have any coefficients in decimal form, convert them into fractions. For example if you have the equation {{{y=.2x^2+.5x+.1}}} convert it into {{{y=(1/5)x^2+(1/2)x+1/10}}}

==section solution
perl

sub replace {
my $exp=$_[0];
$exp=~s/\//z/g;
$exp=~s/\W+//g;
$exp=~s/z/\//g;
return $exp;
}

sub gcf {
  my ($x, $y) = @_;
  ($x, $y) = ($y, $x % $y) while $y;
  return $x;
}

sub lcm {
  return($_[0] * $_[1] / gcf($_[0], $_[1]));
}

sub multigcf {
  my $x = shift;
  $x = gcf($x, shift) while @_;
  return $x;
}

sub multilcm {
  my $x = shift;
  $x = lcm($x, shift) while @_;
  return $x;
}

sub mult{

my (@arr1,@arr2);

@arr1=($_[0],$_[1]);
@arr2=($_[2],$_[3]);

if(@_==2)
{@arr1=split(/\//,$_[0]);
push(@arr1,1) if @arr1==1;
@arr2=split(/\//,$_[1]);
push(@arr2,1) if @arr2==1;
}

return "".($arr1[0]*$arr2[0])."/".($arr1[1]*$arr2[1])."";

}

sub add{

my @arr1=($_[0],$_[1]);
my @arr2=($_[2],$_[3]);

if(@_==2)
{@arr1=split(/\//,$_[0]);
push(@arr1,1) if @arr1==1;
@arr2=split(/\//,$_[1]);
push(@arr2,1) if @arr2==1;
}

if(($arr1[1]!=$arr2[1])&&(($arr1[1]!=0)&&($arr2[1]!=0)))
{
    my $gcf=gcf($arr1[0],$arr1[1]);

my $lcm=($arr1[1]*$arr2[1])/$gcf;

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

$arr1[0]=($lcm/$arr1[1])*$arr1[0];
    $arr2[0]=($lcm/$arr2[1])*$arr2[0];

$arr1[1]=($lcm/$arr1[1])*$arr1[1];
    $arr2[1]=($lcm/$arr2[1])*$arr2[1];
}
$arr1[0]+=$arr2[0];

return "$arr1[0]/$arr1[1]";
}

sub subtract{

my @arr1=($_[0],$_[1]);
my @arr2=($_[2],$_[3]);

if(@_==2)
{@arr1=split(/\//,$_[0]);
push(@arr1,1) if @arr1==1;
@arr2=split(/\//,$_[1]);
push(@arr2,1) if @arr2==1;
}

my $return=add($arr1[0],$arr1[1],-$arr2[0],$arr2[1]);
return $return;

}

sub reduce{

my @arr1=@_;
my @original_arr1=@arr1;

if(@_==1)
{@arr1=split(/\//,$_[0]);
push(@arr1,1) if @arr1==1;
}

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

if($arr1[1]!=0)
{
  my $gcf=gcf($arr1[0],$arr1[1]);
  $arr1[0]/=$gcf;
  $arr1[1]/=$gcf;
}

return "$arr1[0]/$arr1[1]";
}

sub invert{

my @arr1=@_;

if(@_==1)
{@arr1=split(/\//,$_[0]);
push(@arr1,1) if @arr1==1;
}

return "$arr1[1]/$arr1[0]";
}

sub divide{

my @arr1=($_[0],$_[1]);
my @arr2=($_[2],$_[3]);

if(@_==2)
{@arr1=split(/\//,$_[0]);
push(@arr1,1) if @arr1==1;
@arr2=split(/\//,$_[1]);
push(@arr2,1) if @arr2==1;
}

my $return=mult($arr1[0],$arr1[1],$arr2[1],$arr2[0]);
return $return;
}

my $graph_dimensions="500,500,-10,10,-10,10";

$a=$my_a;
$b=$my_b;
$c=$my_c;

if($a==0)
{print "
<font size=\"4\" color=\"red\"><b>Error:</b></font> the leading coefficient \"a\" cannot be equal to zero.

";
return;
}

$a=~s/\)|$//g;
$b=~s/$|$//g;
$c=~s/$|\(//g;

my (@a,@b,@c);

@a=split(/\//,$a);
@b=split(/\//,$b);
@c=split(/\//,$c);

$a[1]=1 if @a==1;
$b[1]=1 if @b==1;
$c[1]=1 if @c==1;

#my ($displayed_a,$displayed_b,$displayed_c)=($a,$b,$c);
#$displayed_a="($displayed_a)" if $displayed_a=~m/\//;
#$displayed_b="($displayed_b)" if $displayed_b=~m/\//;

$a="($a)" if $a=~m/\//;
$b="($b)" if $b=~m/\//;

my ($inner_expression,$factored_b,$factored_c,$display);

#$original_equation="$a x^2+$b x+$c";
 $display="y=$a x^2+$b x+$c";
 $display=~s/\+\-/\-/g;
 my $original_equation=$display;
 $original_equation=~s/y=//;

print "

{{{$display}}} Start with the given equation

my (@factored_b,@factored_c,$abs);

$factored_b=divide(@b,@a);
@factored_b=split(/\//,$factored_b);
$factored_b=reduce(@factored_b);
@factored_b=split(/\//,$factored_b);

$factored_b=~s/\/1$//;

#my @factored_c=divide(@c,@a);
#$factored_c=reduce(@factored_c);
#@factored_c=split(/\//,$factored_c);

$inner_expression.="x^2+";

$inner_expression.="(".join("/",@factored_b).")x+" if $factored_b[1]!=1;
$inner_expression.="$factored_b[0]x+" if $factored_b[1]==1;
#$inner_expression.="".join("/",@factored_c)."+" if $factored_c[1]!=1;
#$inner_expression.="$factored_c[0]+" if $factored_c[1]==1;

$inner_expression=~s/\+$//;

my $negative_c="-$c";
$negative_c=~s/\-\-/\+/g;

if(length($c)!=0)
{$abs=$c;
 $abs=~s/^\-//;
 $display="y$negative_c=$a x^2+$b x";
 $display=~s/\+\-/\-/g;
 print "

{{{$display}}} ", ($c=~m/\-/) ? "Add {{{$abs}}} to" : " Subtract {{{$abs}}} from";

print " both sides

";}

#if($a!=1)
#{
 $display="y$negative_c=$a($inner_expression)";
 $display=~s/\+\-/\-/g;
print "

{{{$display}}} Factor out the leading coefficient {{{$a}}}

";
#}

my $half_b=divide(@factored_b,2,1);
my @half_b=split(/\//,$half_b);
$half_b=reduce(@half_b);
@half_b=reduce($half_b);

$half_b=~s/\/1$//;

my $square_b=mult(@half_b,@half_b);

my @square_b=split(/\//,$square_b);
$square_b=reduce(@square_b);
@square_b=reduce($square_b);

$square_b=~s/\/1$//;

my $before_factor=$inner_expression;
$before_factor.="+$square_b";
$inner_expression.="+$square_b-$square_b";

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

my $after_factor="x+$half_b";
$after_factor=~s/\+\-/\-/g;

print "

Take half of the x coefficient {{{$factored_b}}} to get {{{$half_b}}} (ie {{{(1/2)($factored_b)=$half_b}}}).

Now square {{{$half_b}}} to get {{{$square_b}}} (ie {{{($half_b)^2=($half_b)($half_b)=$square_b}}})

$display="y$negative_c=$a($inner_expression)";
$display=~s/\+\-/\-/g;

print "

{{{$display}}} Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of {{{$square_b}}} does not change the equation

$display="y$negative_c=$a(($after_factor)^2-$square_b)";
$display=~s/\+\-/\-/g;

print "

{{{$display}}} Now factor {{{$before_factor}}} to get {{{($after_factor)^2}}}

$display="y$negative_c=$a($after_factor)^2-$a($square_b)";
$display=~s/\+\-/\-/g;
$display=~s/\-\-/\+/g;

@square_b=split(/\//,$square_b);
$square_b[1]=1 if @square_b==1;
my $new_c=mult(@square_b,@a);
my @new_c=split(/\//,$new_c);
$new_c=reduce(@new_c);

$new_c=~s/\/1$//;

print "

{{{$display}}} Distribute

$display="y$negative_c=$a($after_factor)^2-$new_c";
$display=~s/\+\-/\-/g;
$display=~s/\-\-/\+/g;

print "

{{{$display}}} Multiply

### Optional

$abs=$negative_c;
$abs=~s/\-//;

$display="y=$a($after_factor)^2-$new_c+$c";
$display=~s/\+\-/\-/g;
$display=~s/\-\-/\+/g;

my $final_c=subtract(@c,@new_c);
my @final_c=split(/\//,$final_c);
$final_c=reduce(@final_c);

$final_c=~s/\/1$//;
print "

{{{$display}}} Now add {{{$abs}}} to both sides to isolate y

$display="y=$a($after_factor)^2+$final_c";
$display=~s/\+\-/\-/g;
$display=~s/\-\-/\+/g;

print "

{{{$display}}} Combine like terms

my $final_equation=$display;
$final_equation=~s/y=//;

$a=~s/\)|\(//g;
my $h="-$half_b";
$h=~s/\-\-//;

my $k=$final_c;

print "

Now the quadratic is in vertex form {{{y=a(x-h)^2+k}}} where {{{a=$a}}}, {{{h=$h}}}, and {{{k=$k}}}. Remember (h,k) is the vertex and \"a\" is the stretch/compression factor.

$original_equation=~s/\s//g;

print "

Check:

Notice if we graph the original equation {{{y=$original_equation}}} we get:

{{{graph($graph_dimensions,$original_equation)}}} Graph of {{{y=$original_equation}}}. Notice how the vertex is ({{{$h}}},{{{$k}}}).

Notice if we graph the final equation {{{y=$final_equation}}} we get:

{{{graph($graph_dimensions,$final_equation)}}} Graph of {{{y=$final_equation}}}. Notice how the vertex is also ({{{$h}}},{{{$k}}}).

So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.

==section output
a

==section check
a b c h k