Question 108982
There was a party and everyone shook everyone's hands. There were 66 handshakes!! How many people were at the party?

 
we know that:

with two people (x and y), there is {{{one}}} handshake; ({{{x}}} with {{{y}}})

with three people (x,y, and z), there are {{{three}}} handshakes; x with y and z, and y with z


with four people (x, y, z, and e), there are {{{six}}} handshakes; x with y, z, and e, then  y with z and e, then  z with e

In general:

 with {{{n+1}}} people, the number of handshakes {{{is -the- sum}}} of the

 first {{{n}}} consecutive numbers: {{{1+2+3+ ... + n}}}

Since this {{{sum}}} is {{{n(n+1)/2}}}, 

we need to solve the equation {{{n

(n+1)/2 = 66}}} ...multiply both sides by {{{2}}}.=> {{{n(n + 1)= 132}}}....=>...{{{n^2 + n = 132}}}

this is the quadratic equation {{{n^2+ n -132 = 0}}} 

use quadratic formula and solve for {{{n}}}:

{{{x[1,2]=(-b +- sqrt (b^2 -4*a*c )) / (2*a)}}}

since {{{a = 1}}}}, {{{b = 1}}}, and {{{c = -132}}}, you will have:

{{{x[1,2]=(-1 +- sqrt (1^2 -4*1*(-132) )) / (2*1)}}}

{{{x[1,2]=(-1 +- sqrt (1 + 528 )) / 2}}}

{{{x[1,2]=(-1 +- sqrt (529 )) / 2}}}

{{{x[1,2]=(-1 +- 23 ) / 2}}}

we need only positive root:

{{{x[1]=(-1 + 23 ) / 2}}}

{{{x[1,2]= 22 / 2}}}

{{{x[1,2]= 11}}}


 we obtain {{{11}}} as the answer and deduce that {{{there- were -12- people- at- the- party}}}.