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We  have  a  "low/ high"  situation....so.....this graph has a horizontal asymptote at  y  = 0

So....we need to solve this

  2x - 6

______________  =   0

x^3 -7x^2 -2x + 6

Only  an x value  in the numerator can make this =   0

So

2x  - 6   = 0       add 6 to both sides

2x  = 6  divide both sides  by  2

x =   3

So....it crosses its horizontal asymptote  at  (3, 0)

See the graph here :

https://www.desmos.com/calculator/fotxpapass

LOL!!!....I dunno  about that.....sometimes....I  make  the "easy" ones look "hard"

When CPhill makes a question look SO easy: every looks down...

The winner  will  play

log 1024  / log 4  =  5   games

Everytime, the total number of players is divided by 4

1024 = 4^5 

Answer 5 games

1024    512    256    128    64

oops ....misread the Q !  

n=0;p=0;cycle: a=(100000+n);b=int(a/100000);c=int(a/10000)%10;d=int(a/1000)%10;e=int(a/100)%10;f=int(a/10)%10;g=a%10;n=n+1;if(b==c and b==d and b==e or c==d and c==e and c==f or d==e and d==f and d==g or e==f and e==g and e==b or f==g and f==b and f==c or g==b and g==c and g==d, goto loop,goto cycle);loop:p=p+1;printa,", ",;if(n<=899999, goto cycle, 0);print"Total = ",p

OUTPUT =4,905 such numbers.

Note: Melody, CPhill, EP........Guys, see if you can verify this number mathematically !!. Thanks.

Nice work , Chris !   

For convenience....let there  be  100  children who go to  the doctor.....so...

100 * .70  =    70 have fevers

70 * .30  =  21 have  fevers  and rash

100  - 70  = 30  have no fever

30 * .20  =  6  have  no fever  but  have a rash

21  + 6  =   27  have  rashes

So

6  have  a rash  but  no fever

_______________________  =   6  / 27   =   2 / 9

27    have a rash

EDIT.....thanks, Geno forspotting my earlier error   !!!

All correct.....good job  !!!!

This  will  follow  the principle  that  the angle  of incidence of the light beam  =  the  angle  of  reflection  of the light beam

We can solve this by using similar triangles such  that.....

1.8 /  2.75  =   height  of tower  / 500         cross-multiply

500 * 1.8 / 2.75  = height of tower  ≈  327.3  m

Proof

arctan( 373.3 / 500)  = 33.2°   =  angle of incidence

arctan   ( 1.8/ 2.75)  = 33.2°  = angle of reflection

What big blue box?

We  have  that

pq  = p + q       (1)          

p - q  = 7  ⇒  p   =  7 + q    (2)

Sub  (2)  into (1)  and we have  that

(7 + q)q  =  7 + 2q       simplify

7q  + q^2  = 7 + 2q      rearrange

q^2  + 5q  -  7  =  0   complete  the  square  on q

q^2  + 5q   + (25/4)  =  7  + 25/4

(q + 5/2)^2  =   53/4      take the positive root

q + 5/2  =    √53/2

q =   √53/2 - 5/2    =    [√53 - 5 ] / 2

q^2  =    [ 53 - 10√53 + 25]  / 4   =   [ 78 - 10√53  ] / 4    =   [39 - 5√53] / 2

So....p =  √53/2  - 5/2  + 7    =  √53/2  + 9/2   =   [ √53  + 9 ]  / 2

p^2   =  [ 53  + 18√53  + 81] / 4   =  [ 134 +18√53 ] / 4  =  [ 67 + 9√53  ] / 2

Simplifying  the given  expression  we   have  that

(pq)^2                 (p + q)^2         

_________ =     ________ 

p^2 + q^2           p^2 + q^2

p + q  =   √53 + 2

(p+ q)^2   = (  53  + 4√53  + 4)   =  (57 + 4√53)

p^2  + q^2    =    [106  + 4√53 ] / 2   =  53  + 2√53

So  we  have  

(57 + 4√53 )        (53  - 2√53)             2597  + 98√53       

___________     ___________  =    ______________

( 53 + 2√53)        (53  - 2√53)               2597

So

a +  b  +  c   +  d  = 

2597  + 98 + 53   +  2597   =    5345

6370 km  is any of the following:

637.0 x 10^1

63.70 x 10^2

6.370 x 10^3  km    (NOT kilos)      the positive exponent tells how far to move the decimal point to the right.

DUDE THX

sqrt ((x+3)^2)  = 7    square both sides

(x+3)^2  = 49            

x+3    = +-7

x = 4   or  -10    

b

2b + 4

2b + 4 -4b

(2b + 4 - 4b) / 2     =    (4 - 2b)/2 =  2 - b

Looks OK    ,  jjl

The NUMERATOR    factors  into   6y^2 (2y-1) ........does that tell ya anything?  

Looks good to me , jjl.......but I do not know what all that stuff in the big blue box is about.....

M = 1/2 L -5     (given)

(L-9)   - (M-9) = 21     sub in the 'given' for M

L-9  - (1/2L-5  -9) = 21

1/2L  -9 +5+9 = 21

1/2 L = 16

L=32

                  then    M = 1/2 L -5  =11 y/o