Melody

Let a and b be real numbers such that the equation 6x^2 - axy - 3y^2 - 24x + 3y + b = 0 represents two lines, whose intersection lies on the x -axis. Enter the ordered pair (a,b). 

If the intersection is on the x axis then that is when y=0

sub in y=0 and I get

\(6x^2 - axy - 3y^2 - 24x + 3y + b = 0\\ when\;\;y=0\\ 6x^2 - 24x + b = 0\\ x=\frac{24\pm\sqrt{24^2-24b}}{12}\\ \text{if there is only one solution then}\\ 24^2-24b=0\\ 24-b=0\\ b=24\)

if b=24 then we have

\(6x^2 - axy - 3y^2 - 24x + 3y + 24 = 0\)

I have used a Desmos graph to determine that a=1.5 approx

So    (a,b) = (1.5,24)        

BUT I have not worked out how to get the value of a without using the graphing calculator.

My knowledge of conics is not great.   

The function f satisfies

f(sqrt(x+1)
for all x>=-1, x isn't equal to 0. Find f(2).

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This notation looks odd to me. There is a bracket messing but even with that it does not make a lot of sense to me.

Could you check it please?

All I can say is that  if  you want f(sqrt(x+1))  and sqrt(x+1)=2 then

\(\sqrt{x+1}=2\\ x+1=4\\ x=3\)

I cannot find f(2) becasue the function is not defined. There are no equal signs.

asad -  did you actually try to answer this. If you did I doubt you were sucessful.  

If you want to give people hints that is fine but if you have not fully answered the question first I think you should also admit that.

I know in the past I have had people who have given me hints when they do not know how to do it themselves.

I have assumed that they do know how to do it and it can be very frustrating.

If they had admitted that they did not know but they were just trying to brainstorm with me then that would have been really good.

Here are my thoughts - I did get a different answer from you Rom :/

How many students at the school play on exactly two teams?

At a school, all 60 students play on at least one of three teams: Basketball, Soccer, and Mathletics.

8 students play all three sports,       So 8 can be put where all the circles overlap.

half the students play basketball,      So 30 do basketball.  That is why I put 30 in a box for basketball.

maths:basketball:soccer    is 4:3:2 . 

So we have ..  

From the Venn diagram we can say

x+y+z+2(a+b+c)+3*8=90    (1)    and

x+y+z+   a+b+c   +8  =60    (2)

subtract

a+b+c+2+8 = 30

a+b+c = 14

So I think that 14 students do exactly 2 sports.

I do not know what the answer is but I did find this artical.

Hi Jess,

I'd really like to see someone answer this question too!   

PM it is not 1/1 

The top has + signs and the bottom has - signs. So the ansswer is more than 1

Your question is not complete. The numbers are missing.

The volume of any right pyramid is    1/3  *  the area of the base.

Actually I am not sure if it has to be a right pyramid, I think it can be any pyramid.

Tetre got it from (3x)^2 = 9x^2   Thanks Tertre and asdf   

There is an obscure circle geometry  law that says if A is a point external to the circle

then the square of the tangential distance from A to the circle 

is equal to the square of the distance from A to the furthest point on any chord that is collinear with A.

Maybe that can be worded better. :/

anyway it says     \((3x)^2=(9+16)^2\)

No one on this post has attempted to prove this.  They have simply used it to answer your question.