Remember that the formula for the number of handshakes possible at a party with n people is:
# handshakes = n*(n - 1)/2
So, if given the number of handshakes, it is possible to determine how many people are at the party. Suppose there are 105 handshakes. Then, using the formula, we know
105 = n*(n - 1)/2
Multiplying both sides by 2, we get
210 = n*(n - 1)
210 = n2 - n
0 = n2 - n - 210
Notice that this is a quadratic equation. So, it can be solved by factoring or the quadratic formula. Here, we'll factor the equation.
0 = (n - 15)(n + 14)
So, n = 15 or -14. However, since there cannot be a negative number of people, the number of people at the party must be 15.
Sometimes you’re going to be asked how many people are at party and you know the number of handshakes. So here we know that there is 105 possible handshakes at a party. Let’s start by writing down our equation. We know that the number of handshakes, I’ve abbreviated hs, is equal to the number of people times the number of people minus 1, all divided by 2.
So which do we know, do we know the number of people or do we know the number of handshakes? Well if I read the problem again I see that there are 105 handshakes, yet again rereading a problem is okay it doesn’t mean that you’re dumb if you have to reread a problem. So we’re going to have 105 equals n times n minus 1. I was tempted to distribute there and I know that I want to do one step per line. So if we solve this for n we’ll know the number people.
So I’m going to start by multiplying by 2, 2 times 105 is 210 and that equals n times n minus 1. I’m going to distribute that n, so now we have 210 is equal to n² minus n. To solve this quadratic I need a 0 on the left side, so I’m going to subtract 210 and I see that I'm running to the bottom here so I’m going to move back up. So we have 0 is equal to n² minus n minus 210. You could use the quadratic formula, however I think factoring and using the zero product property is a little easier.
N times n will give you n², multiplies to a negative adds to a positive we know that our negative number’s larger. N 210 is 15 less than 225 which is 15². So I see that n plus 14, n minus 15 are two binomials. Using the zero product property here I can say that if n is -14 that binomial equals zero. Using the zero product property here I can say that n equals 15. But the questions is, which of those two answers are we going to use. Well you don’t have negative people if they’re dull, we’re going to have positive 15 people. So that’s our answer we have 15 people which means there are 105 possible handshakes at a party.