There are n sweets in a bag, 6 orange, rest yellow. H takes two, one after another, and eats them. Probability both are orange is 1/3. Show n^2 - n - 90 = 0.

First, we start with what we do know and this should lead us to the answer with steps inbetween. So the amount of yellow sweets to start is (n-6) but we want to focus on the orange sweets. This question is asking about probability. We are given a probability and so we shall write the equation in terms of n for this, as the equation we want is in terms of n too. So probability of her eating two orange sweets is first the probability of eating an orange sweet the first time, 6/n which is 6 sweets out of n sweets, and then multiplying this by the probability of eating a orange sweet the second time, this is 5/(n-1) as we have one orange sweet taken out the last time so 6-1=5 and we lost a sweet so overall number of sweets is (n-1). Then (6/n)x(5/(n-1)) = 1/3. We can see 90 can be formed from 6x5x3 so it is looking like a useful equation for us. So we need to just play around with the numbers and multiply all the denominators up, so multiply by 3n(n-1). 90=n(n-1)=n^2 - n. So n^2 - n - 90 = 0, which is what we want to show.