ElectricPavlov

y = 1 (x) 

find perpendicular line 

y = -x 

that passes through the point (5,7) 

y = -x + 12 

Intersection point is 6,6     ( found by equating    y =  x   and   y = -x +12 ) 

(6,6) IS THE midpoint BETWEEN the original point (5,7) and the reflected point (x,y)  

    so      ( 5+x ) / 2 = 6    and    (7+y) / 2 = 6    shows  (x,y) to be    (7, 5) 

This is pretty easy to see with a line  y = x .....but this is the method to use for more diffiicult line equatins such as ' y = 5x '    etc......

There are 2 Americans ...they can be either 1st 2nd 3rd =   6 ways 

   the 5 Canadians fill one of the two remaining spots = 10 ways                  10 x 6 = 60 ways for ONE cnadian and ONE American to medal together 

        the final spot is then for one of the remaining 12 

                12 x 60 = 720 ways 

If a Canadian or American can win the third spot , there are 17 skaters that can fill the third spot 

               17 x 60 = 1020 ways 

(I think) 

x^2 - 10x + y^2 + 10y + c = 0      Arange to standard circle form of an equation

x^2 -10x  + 25      + y^2 +10y + 25   =  -c  + 25 + 25 

(x-5)^2      +   (y+5)^2   = - c +50               circle centered at  5, -5    with radius^2 =  - c + 50 

                                                                                      if radius = 1    radius^1 = 1       then     - c + 50 = 1        c = 49 

First, simplify to :

3x^2 -15x - 90      then manipulate ot vertex form 

3 ( x^2 - 5x)   - 90 

3 ( x- 2.5)^2    - 18.75 - 90 

3 (x-2.5)^2 - 108.75                            <===== shows vertex at   2.5 , -108.75        n = - 108.75

AI approval is a bit 'sucky' at present, so I have no idea what was posted for an answer....here is my answer:

In quadratic form 

x^2 + (k-9) x + 16        <====== determinant of this quadratic must = 0 for only one root

b^2 - 4 ac = 0 

(k-9)^2 -4(1)(16) = 0 

   k^2 - 18k + 17 = 0 

     ( k -1)(k-17)   = 0               shows k = 1 or 17     sum = 18 

(x -4)^2  = x^2 - 8x + 16       ✔

(x+4)^2  = x^2 +8x + 16        ✔            Check!

Job / 5     = number of worker_days required

job / ( 5 +1)  = job/5  - 8 

job = 6 job/ 5 - 48 

-1/5 job = -48 

job = 240 worker_days   <==== so it takes   5 workers   48 days 

                 to complete in 20 days ( 28 days earlier) 

                        240 worker _ days / x = 20 

                             x = 12 workers required      you will need to hire an dditional 7 workers over the original 5 

From the diagram below, using Pythagorean theorem:

2.5^2 + (5-r)^2  = (2.5+r)^2 

6.25 + 25 - 10r + r^2 = 6.25 + 5 r + r^2 

-15r = -25 

r = 1  2/3  units             ( and diameter = 2 * r = 3 1/3   units )