Well, the main problem I see is that you are using x instead of * for multiplication. x is the string repeat operator. For example

print 3 x 4;  # This prints "3333"
print 3 * 4;  # This prints 12
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Added Here is a fixed and cleaned up version of your code:

#!/usr/bin/perl -w
use strict;
use Regexp::Common;

my $a = prompt("A? = ");
my $b = prompt("B? = ");
my $c = prompt("C? = ");

my $root = sqrt(($b**2) - 4 * $a * $c) / (2 * $a);
my $ans1 = -$b + $root;
my $ans2 = -$b - $root;

print "Answers: $ans1 AND $ans2\n\n";

sub prompt {
    my $var;
    do {
        print $_[0];
        $var = <STDIN>;
        chomp $var;
    } while ($var !~ /\A$RE{num}{real}\z/);
    return $var;
}
[download]

The major thing I added was the prompt sub. It will prompt the user for a value, and if the value isn't a real number, it will continue to prompt them until they enter a real number.

And as for a more elegant way of making a number negative, simply do -$b.