Question 176375
Call the number of people currently in the club x.  Then if they share the cost equally, they will each pay $200,000/x.  Now, if they find 5 more people the cost will decrease by 2,000 per person.  So if x is increased by 5, (x+5), the cost, $200,000/(x+5) will be 2,000 less per person, a.k.a. $200,000/x - 2000.  So we need to set those two equal to each other and solve for x.
{{{200000/(x+5)=200000/x-2000}}}
Now, if you are anything like most people, than I would assume you don't like working with fractions.  So how about we try to get rid of them?  If we multiplied both sides by x would that work?  No, it would not get rid of the fraction on the left hand side.  If we multiplied both sides by x+5 would that work?  No again, as it would not correct the fraction on the right hand side.  It seems that we need both.  So let's multiply by x(x+5) on both sides:
{{{200000x=200000(x+5)-2000x(x+5)}}}
{{{200000x=200000x+1000000-2000x^2-10000x}}}
Now, subtract 200000x from both sides to obtain:
{{{0=1000000-2000x^2-10000x}}}
Rearranging, we have a quadratic:
{{{-2000x^2-10000x+1000000=0}}}
Now, these are all fairly large numbers.  Do they have anything in common?  They all have at least a factor of 2000 in them, so let's divide both sides by -2000 to get rid of it:
{{{x^2+5x-500=0}}}
So now, we need to solve this quadratic equation.  I will use the quadratic formula:
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
Here, a=1, b=5, c=-500, so
{{{x = (-5 +- sqrt( 5^2-4*1*-500 ))/(2*1) }}}
{{{x = (-5 +- sqrt(25+2000))/2}}}
{{{x = (-5 +- sqrt(2025))/2}}}
{{{x = (-5 +- 45)/2}}}
So,
{{{x=40/2=20}}}
and
{{{x=-50/2=-25}}}
However, it's not possible to have -25 people in a club, so the only feasible solution here is 20.