Solver Min/Max of a Quadratic Function

Source code of 'Min/Max of a Quadratic Function'

This Solver (Min/Max of a Quadratic Function) was created by by hummingbird(0)

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==section input

Find min/max *[input v=x] this equation: *[input a=1]x2+*[input b=-1]x+*[input c=-2] = 0. You can put negative numbers if you need to use a negative coefficient. The variable that you specify will be used instead of x.

==section solution
perl

if( $a == 0 ) {
  print "Come on, this is not a quadratic equation!\n";
  return;
}

print "The min/max of a quadratic equation is always at a point where its first differential is zero. This means that in our case, the value of {{{$v}}} at which the given equation has a <A HREF=Maxima_and_minima.wikipedia>maxima/minima</A> must satisfy the following equation:";

my $soln = -$b/(2*$a) ;
my $mm = "maxima"; 
my $x=(-$b)/2;
if($a>0)
{  
   $mm = "minima";
}

print "{{{2*$a*$v+$b=0}}}
=> {{{$v = -$b/(2*$a) = $soln}}}

This point is a minima if value of coefficient of $v2 is positive and vice versa. For our function the point $v=$soln is a {{{$mm}}} ";

print "The graph of the equation is :\n\n{{{graph( 500, 500, -10, 10, -20, 20, $a*x^2+$b*x+$c )}}}";

print "
Alternate method

In this method, we will use the <A HREF=Perfect_square.wikipedia>perfect square</A> method.

Step one:
Make the coefficient of {{{x^2}}} positive by multiplying it by {{{-1}}} in case{{{a<0}}}.
Maxima / Minima is decided from the sign of 'a'. 
If 'a' is positive then we have Minima and for 'a'negative we have Maxima.";

my $x=(-$b)/2;
if($a>0)
{  
   $V=$v;
   $B=$b;
   $a=$a;
}
else
{
   $V=-$v;
   $B=-$b;
   $a=-$a;
}
my $A=sqrt($a);
my $Sol =-($B/2)/$A;
print "

Step two:
Now make the perfect square with the same {{{x^2}}} and {{{x}}} coefficient. ";

print "
{{{($A*x + ($B/2))^2}}}

Maxima / Minima lies at the point where this squared term is equal to zero.

Hence,
=>{{{x=(-($B/2)/$A)= $Sol}}}";
print"

This point is a minima if value of coefficient of $v2 is positive and vice versa. For our function the point $v=$soln is a {{{$mm}}}.

For more on this topic, refer to <A HREF=Quadratic_equation.wikipedia>Min/Max of a Quadratic equation</A>.";
==section output
soln, mm

==section check
  a=1 b=-1 c=3 soln=0.5 mm="minima"