[-
        use Algebra::GenPage;
	use Algebra::SolverLib;
	use Algebra::HTML;
	use Enurl;

-]

[-
	$escmode = 0;

	sub graph_link {
		my ($a, $b, $c) = @_;
		my $encoded = enurl { 'a'=>$a, 'b'=>$b, 'c'=>$c };
		return "\<IMG SRC=quadgraph.mpl?$encoded\>";
	}

	sub reduce_fraction {
		my ($num, $denom) = @_;

		if( int($num) != $num || int($denom) != $denom ) {
			return "$num/$denom";
		}

		if( (int( $num/$denom )) == ($num / $denom) ) {
			return $num / $denom;
		}

		my $n1 = $num < 0 ? -$num : $num;
		my $n2 = $denom < 0 ? -$denom : $denom;
		while( $n1 != $n2 ) {
			if( $n1 > $n2 ) {
				$n1 = $n1 % $n2;
				if( $n1 == 0 ) {
					$n1 = $n2;
					last;
				}
			} else { 
				$n2 = $n2 % $n1;
				if( $n2 == 0 ) {
					$n2 = $n1;
					last;
				}
			}
		}
		return $num/$n1 . "/" . $denom/$n1;
	}

	sub signed {
		my $num = shift @_;
		return "+$num" if $num >= 0;
		return $num; # negative
	}
	sub sign {
		my $num = shift @_;
		return '+' if $num >= 0;
		return '-';
	}
	sub term {
		my $num = shift @_;
	        return $num + 2 - 1 - 1 if $num >= 0;
		return "($num)";
	}


	$a = "$fdat{a}";
	$b = "$fdat{b}";
	$c = "$fdat{c}";

	$a = 1 if $a eq "";
	$b = 1 if $b eq "";
	$c = 0 if $c eq "";

	$first_sign = ($fdat{first_sign} eq '+') ? 1 : -1;
	$signed_b = $b * $first_sign;
	$second_sign = ($fdat{second_sign} eq '+') ? 1 : -1;
	$signed_c = $c * $second_sign;
	$discr = $b*$b-4*$a*$signed_c;
	$ac4=-4*$a*$c*$second_sign;
	if( $a != 0 ) {
		if( $discr > 0 ) {
			$x1 = (-$signed_b + sqrt( $discr ))/2/$a;
			$x2 = (-$signed_b - sqrt( $discr ))/2/$a;
			$solution = "$x1, $x2";
			$factored = ($a != 1 ? $a : "") . "(x". signed(-$x1) .")(x". signed(-$x2) .")";
		} elsif( $discr == 0 ) {
			$solution = -$signed_b/2/$a;
			$factored = ($a != 1 ? $a : "") . "(x". signed(-$solution) .")(x". signed(-$solution) .")";
		} else {
			$solution = "no real solutions";
			$factored = "Expression cannot be factored";
		}
		&process_request( \%fdat, \%udat, "Solve Quadratic Equation $a"."x^2".signed($signed_b)."x".signed( $signed_c )." = 0" );
	} else { 
		$solution = "Sorry , the first coefficient is zero, which makes the equation not a quadratic equation.";
		$factored = "Expression cannot be factored";
	}

	$escmode = 0;
-]

[+ 
	make_page_header( 
		"Quadratic Solver", 
		"Algebra.Com", "Algebra", "Algebra Homework" ) 
+]

<HR>

<B><A HREF=practice.epl>
    [NEW!]Practice with Quadratic Equations! Improve Your Test Scores!</A>
</B>
<BR>
<B><A HREF=http://www.algebra.com/algebra/homework/word/quadratic/index.epl>
	[NEW!]Check out Word Problems involving Quadratics!
</A></B><BR>

[+ lesson_link( "quadform" ) +]


<CENTER><H2> Quick Answer </H2></CENTER>
<TABLE><TR><TD VALIGN=top>
	<CENTER>
		Actual Graph of  y = 
		[+ "$a"."x\<SUP>2\</SUP>" . sign( $signed_b ) . "$b"."x"
			. sign( $signed_c ) . "$c " +]
		<BR>
		[+ graph_link( $a, $signed_b, $signed_c ) +]<BR>
		See [+ lesson_link( "graphing" ) +]
	</CENTER>
</TD><TD VALIGN=TOP>
<UL>
	<LI> <FONT SIZE=+2>x = [+$solution+]</FONT> (sometimes solutions may be close 
	     approximations of the actual solutions)<BR>
	<LI>
		<P>
		Expression factored:

		[+ "$a"."x\<SUP>2\</SUP>" . sign( $signed_b ) . "$b"."x"
			. sign( $signed_c ) . "$c = " +] [+ $factored +]
	<LI> See [+ lesson_link( "quadform" ) +]
</UL>
</TD></TR></TABLE>

<H2> Solution Explained</H2>

Equation: <CODE> [+ plot_formula( "$a"."x^2" . sign( $signed_b ) . "$b"."x". sign( $signed_c ) . "$c = 0" ) +] </CODE>
<P>


<CODE>
a = [+$a+]<BR>
b = [+$signed_b+]<BR>
c = [+$signed_c+]<BR>
</CODE>
<P>

Discriminant: <CODE>b<SUP>2</SUP>-4ac = 
[+ $b +]<SUP>2</SUP>[+ sign( $ac4 ) . "4*" . term( $a ) . "*" . term( $signed_c ) . " = $discr"+]</CODE>

<P>

[$ if $a != 0 $]
[$ if $discr > 0 $]
	Discriminant ([+$discr+]) is greater than zero. The equation has two solutions.<P>
<PRE>

[+ plot_formula( "x = (-b +- sqrt( b^2-4ac ) )/2a" ) +]

or

[+ plot_formula( "x = (-" . term($signed_b) . "+- sqrt( $b^2 " . sign( $ac4 ) . "4*" . term( $a ) . "*" . term( $signed_c ) . " ))/(2*" .term ($a) . ")" ) +]

[$ if( sqrt($discr) == int(sqrt($discr)) ) $]

or

x<SUB>1,2</SUB> = ([+-$signed_b+] &plusmn; [+sqrt( $discr )+]) / [+2*$a+]
or

x<SUB>1</SUB> = [+-$signed_b + sqrt( $discr )+] / [+2*$a+] = [+ $x1 = reduce_fraction( -$signed_b + sqrt( $discr ), 2*$a ) +]
x<SUB>2</SUB> = [+-$signed_b - sqrt( $discr )+] / [+2*$a+] = [+ $x2 = reduce_fraction( -$signed_b - sqrt( $discr ), 2*$a ) +]

or

x<SUB>1,2</SUB> = [+$solution+]

<P>
Equation factored:

[+ $factored +]

[$ else $]

or 

[+ plot_formula( "x = (-" . term($signed_b) . "+- sqrt( $discr ) )/(2*" .term ($a) . ")" ) +]<P>

[$ if( int($signed_b/2/$a) == $signed_b/2/$a || int( $discr/4/$a/$a ) == $discr/4/$a/$a ) $]

Reducing it further, we have 

[+ plot_formula( 
	term( reduce_fraction( -$signed_b, 2*$a ) ) 
	. "+- sqrt( " 
	. reduce_fraction( $discr, 4*$a*$a ) 
	. ")" 
   ) 
+]

[$ endif $]

</PRE>

Since [+ plot_formula( "sqrt( $discr )" ) +] is not an integer number
(it is a so called <B>irrational number</B>, not reducible to fractions
like m/n), further reduction of this expression will not give you an
integer result.

[$ endif $]

</PRE>

[$ elsif $discr == 0 $]
	Discriminant ([+$discr+]) is zero. There is only one solution.

<PRE>
x = -b/2a

or

x = [+-$signed_b+]/(2*[+$a+]) = [+-$signed_b+]/([+2*$a+]) = [+-$signed_b/(2*$a)+]
</PRE>
[$ else $]
	Discriminant ([+$discr+]) is less than zero. No solutions are defined.<P>
	<SMALL>Note: for those of you who study complex numbers, there is
	a complex solution. If you do not know what complex numbers are,
	do not worry about a complex solution and just accept the fact
	that there are no solutions to your problem. You will study
	complex numbers later in your school program (if ever). 
	</SMALL>
[$ endif $]
[$ else $]
	[+ $solution +]
[$ endif $]

<B><A HREF=http://www.algebra.com/algebra/homework/word/quadratic/index.epl>
	[NEW!]Check out Word Problems involving Quadratics!
</A></B><P>

<B>Word Problem Example:</B>

<CENTER><TABLE BORDER=2 WIDTH=80% ><TR><TD><B><I>
        
The length of a hypotenuse of a right triangle is <FONT COLOR=brown>2</FONT> feet more
than the longer leg. The length of the longer leg is <FONT COLOR=brown>7</FONT> feet more
than the lenth of the shorter leg. Find the number of feet in
length of each side of the right triangle.

 
</I></B></TD></TR></TABLE></CENTER><P>
<B>
        <A HREF="/cgi-bin/word-problem.mpl?problem_id=quadratic/02-quadratic.wpm">Solve This Problem Interactively</A>
</B>
/
     <B>
        <A HREF="/cgi-bin/word-problem.mpl?problem_id=quadratic/02-quadratic.wpm&action=customize">Customize This Problem</A>
</B>

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