Melody

Here is the graph

Mmm

If one root of the equation x^2 - x - k is square that of other find the other root . Find the value of K.

Roots are

    \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\\ = {1 \pm \sqrt{1+4k} \over 2}\\ We \;\;want\\ \left({1 - \sqrt{1+4k} \over 2}\right)^2 = {1 + \sqrt{1+4k} \over 2}\\ \)

The right hand side is clearly bigger than 1 so the left side (before it is squared) must be less than 0

\(1-\sqrt{1+4k}<0\\ \sqrt{1+4k}>1\\ 1+4k>1\\ 4k>0\\ k>0\)

\( \left({1 - \sqrt{1+4k} \over 2}\right)^2 = {1 + \sqrt{1+4k} \over 2}\\ {1+1+4k - 2\sqrt{1+4k} \over 4} = {2 +2 \sqrt{1+4k} \over 4}\\ 2+4k-2\sqrt{1+4k}=2 +2 \sqrt{1+4k}\\ 2k-\sqrt{1+4k}= \sqrt{1+4k}\\ 2k= 2\sqrt{1+4k}\\ k= \sqrt{1+4k}\\ k^2=1+4k\\ k^2-4k-1=0\\ k=\frac{4\pm\sqrt{16+4}}{2}=2\pm\sqrt5\)

\(\\but\;\; k>0\\ so\\ k=2+\sqrt5\)

So there are 5 girls and 3 dresses. 

Each show has 3 girls wearing a dress each.

I'm a bit unclear on the rules.

I guess one or 2 girls can wear the same dress but not all three.

I think it is the same as how many ways can 3 girls be chosen from 5 if order counts.

Order counts becasue the first one gets the first dress, the 2nd gets the 2nd dress and the 3rd gets the 3rd dress.

So that would be      5P3    =  60 

So I think there are 60 shows. Some of the girls and outfits will be the same but never all 3.

Do you have a picture to go with this description?

It doesn't make a lot of sense to me.

\({x}^{{x}^{{x}^{...}}} = 2 \qquad\\ so\\ x^2=2\\ so\\ x=+\sqrt{2}\)

Thanks Daisy,   

Here is just a little more explanation ... I am doing it the long way but it might be easier to understand ... maybe 

A bat and a ball cost one dollar and ten cents in total. The bat costs a dollar more than the ball. How much does the ball cost?

I am going to change the bat to a racket. So I have  R and B

B+100 = R       all in cents,    I can rearrange this to

B-R     =   -100     (1)

I also know that

B+R    =    110     (2)

Now solve simultaneously  -   add

2B+0   =  10

Ball =    5c

I agree with the 2 guests.

I believe that the answer is   1/35

I did it the same way as the first guest.   

Ginger you did not take into consideration the number of permutaions for each set of outcomes. You have to think of the 5 dice as all being different from each other.

What is the probability that when we roll 5 fair 6-sided dice, at most 4 of them will show 1?

So there can be 0, 1, 

1- P( all ones)

If all the dice are different then there are  6^5  possible permutations.

There is only 1 way that they can all be 1s

so prob = (1-  1/6^5) = (6^5-1)/6^5 

(6^5-1)/6^5 = 7775/7776

I can be one of the 2 hungry people who knows where the glass is so I am not going to eat one of those. 

I guess this is not the answer that you are hoping for :)

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How about.

Let the 2 who know each chose a good peice, then allow them to eat the peice they chose. They are hungry so it they really have chosen a good peice they will eat it and they were telling the truth. If they don't eat it it is because there is glass in that slice and they are lying.

If they both eat it then you can just ask them to sort the good from the bad, they are both telling the truth so they will agree on the out come and the 3 remaining good pieces will be identified.

If neither eat it then they are both liars. You can get them to sort out the 5 good and 5 bad ones 

If neither eat it then you can again ask them to sort the good from the bad, they are both lying so they will agree on the out come.  The 5 that they say are good are the bad ones

If one eats the cake and one doesn't then it is obvious that the eater is telling the truth and the onther is lying.

Get the truth teller to point out the other 4 good ones. 

Yes x=7 but  hectictar I have no idea why you say that x is the solution to this .....

B)     2l x-12 l = 10  

Ther is no l in the question ???