Show that n²-n-90=0

FULL QUESTION: here are n sweets in a bag. Six of the sweets are orange. The rest of the sweets are yellow. Hannah takes a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is 1/3. Show that n²-n-90=0

(This a question from the 2015 Maths GCSE exam which 90% of the students found very hard) The key to answering this question is the think logically.

~There are a total of n sweets ~We are told that 6 of them are orange and the rest are yellow ~therefore n-6 of them are yellow ~If hannah takes a sweet from the bag and that sweet is orange the probability of that is simply: 6/n

now one orange sweet has been consumed so

~The total number of sweets left is : n-1 ~And the total number of orange sweets left is: 5 ~If she reaches for another sweet, the probability of that being orange is: 5/n-1

Now we have two independent probabilities. In this case we have been told that the probability of her getting two orange sweets is: 1/3 Therefore as rule when we multiply two individual probabilities we get the probability for those those events happening together.

Therefore we multiply both probablilties which equals to 1/3 6/n x 5/(n-1) = 1/3 30/n^2-n =1/3 90 = n^2-n n^2-n-90=0 n^2-10n+9n-90=0 n(n-10)+9(n-10)=0 (n+9)(n-10)=0

Solving this we get two answers: 10 & -9 We know the answer is 10 because we can't have a negative amount of sweets

When we put the value we found (10) in the equation n^2-n-90 we get 0 which is exactly what we had to prove!

There are n sweets in a bag. Six of the sweets are orange. The rest of the sweets are yellow.

Hannah takes a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet.

The probability that Hannah eats two orange sweets is 1/3. Show that n²-n-90=0