CPhill

Rewrite

x^2  - 20x =   -64   complete the square on x

x^2 -20x + 100  = -64 + 100

(x -10)^2  = 36

x -10  = sqrt 36               or      x -10   = -sqrt 36

x  -10  = 6                                  x  -10   =  - 6

x = 16                                           x =  4       

[ 15 - 16i ]          [1 + i ]            [ 15 + 15i - 16i + 16 ]          [  31 - i  ]          

________  *    _______  =      _________________  =   _________  =    (31/2) - (1/2) i

     1 - i               [1 + i ]              [  1   - i^2  ]                            2

 (1 - 3i)(2 + i)(4 - 7i)(11 + 13i)

(1-3i) (2 + i)  =  2 -6i + i - 3i^2  =   2 - 5i + 3  =  5 - 5i  =  5 ( 1 - i)

(4 -7i) (11 + 13i)  =  44 - 77i + 52i - 91i^2  = 145 - 25i = 5 (27 - 5i)

5 ( 1 - i) * 5 (27 - 5i)  =

25 ( 27 -27i - 5i + 5i^2)   =  25 ( 22 - 32i) =  550 - 800i 

Simplify as

m + 11n - mn  =  -17

mn - m - 11n  =  17

m ( n - 1)  =     17 + 11n

m =      [17 + 11n]  / [ n  - 1 ] 

n          m

29        12 ( as  n → nfinity , m→11...12 is the smallest integer value for m in this direction) )

15        13

8          15

5          18

3          25

2          39

0         -17

-1         -3

-3          4

-6          7

-13        9

-27       10  ( as n → -infinity, m →  11.....10 is the largest integer value for m in this direction )

\(\dfrac{1}{\sqrt2+2+1-\sqrt{3}}+\dfrac{1}{\sqrt2-2-1+\sqrt{3}} \)

Simplify as

             1                                             1

_________________     +        _________________

sqrt (2) - sqrt (3) + 3                  sqrt (2) + sqrt (3) - 3

  First fraction

1   [  (sqrt (2) - sqrt (3))  -3]                                           sqrt (2) - sqrt (3)   - 3

____________________________________ =   ________________________  =

[(sqrt 2 - sqrt 3) + 3 ] [ (sqrt 2 - sqrt 3) -3 ]            [ (sqrt (2) - sqrt(3)] ^2 - 9]

sqrt (2) - sqrt (3) - 3                     ( sqrt 2  - sqrt 3)  - 3               3  - (sqrt 2 - sqrt 3)

_________________  =             __________________  =       __________________ 

2 - 2sqrt 6 + 3  - 9                               -4  - sqrt 24                                 sqrt (24) + 4

Second fraction 

      1 [ (sqrt 2 + sqrt 3) + 3]                                     3 +   (sqrt 2 + sqrt 3) 

__________________________________   =      ____________________ =

[(sqrt 2 + sqrt 3)  -  3] [ (sqrt 2 + sqrt 3) + 3 ]            (sqrt 2 + sqrt 3)^2  - 9   

   3 + (sqrt 2 + sqrt 3)                          3 +  (sqrt 2 + sqrt 3) 

_______________________  =        _________________

   2 + 2sqrt 6 + 3  - 9                             sqrt (24) - 4

Combine both

[ 3 - (sqrt 2 -sqrt 3)]                     [ 3 + (sqrt2 + sqrt3) ]

___________________     +  ____________________   

       sqrt (24) + 4                                 sqrt (24) - 4

(multiply top / bottom by denominator  conjugate)

[ 3 -sqrt 2 + sqrt 3] (sqrt (24)  - 4 )  + [ 3 + sqrt 2 + sqrt 3 ] [ sqrt (24) + 4 ]

___________________________________________________________

                                    24  - 16

6sqrt (24)  +  2sqrt (72) + 8sqrt 2

___________________________   =

                         8

12sqrt 6  + 12sqrt 2 + 8sqrt 2

________________________ =

                        8

3sqrt 6  +  5sqrt 2

________________

               2

Let O be the center of the quarter-circle

Let M    be the intersection point of the edges of  both circles

Draw  a tangent line  to  the quarter circle at M and  let N be any other point on this line

Angle  OMN = 90°

Let P be the center of the small circle

Angle PMN  also   = 90°   and PM   must be the radius of the smaller circle since it forms a 90° with the tangent line

Then  PM  must lie  on  OM  since angle PMN  = angle OMN

Let S be the other intersection of the radius of the quarter circle and  the eadge of the  smaller circle

PS =  radius of the small circle = r

OP  =  OM - PN  =   4 - r

OS = 2 + r

Triangle  OPS  forms a right triangle such that

OP^2  =  PS^2  + OS^2

(4 - r)^2  =  r^2  + ( 2 + r)^2

r^2 - 8r + 16 =  r^2  + r^2 + 4r + 4

r^2  + 12r - 12  =  0

r^2 + 12r  =  12                complete the square on r

r^2 + 12r + 36   = 12 + 36

(r + 6)^2  =  48         take the positive root

r + 6   = sqrt (48)

r = sqrt (48)  -  6

r = 4sqrt (3)  - 6    ≈  .923

The hexagon is composed of 6 equilateral triangles each with sides of 3 and with an area =  

(1/2)(3^2)sqrt (3)  / 2  =   (9/4)sqrt 3

So......the total area =  6 (9/4) sqrt 3  =  (27/2)sqrt 3  ≈  23.38

Let X  be the point of tangency  to the edges of  both circles  

Let the other two tangent points at the intersection points of the radius of the quarter-circle and the edge of the small circle be Y and Z

Let the center of the small circle  = M

Connect  MY , MZ and MO

Triangles MYO  and MZO  form congruent  45 - 45 - 90 right triangles

Let the radius of the small circle  = r

MZ =  r     MO =  (sqrt2) * r

Connect OX

OX  = MO  +  MX

OX =  2       MO = sqrt2 * r      MX  = r

So

MO + MX  = OX

sqrt (2)r + r =  2

r ( sqrt 2  + 1)  =  2

r =  2 / (sqrt (2)   + 1)  ≈  .828

Let d be the common difference  between the angle measures

Let a  be the smallest angle and  a + 56   = the largest  =  a + 3d

The sum of the  interior angles =  360

So

a + 3d  =  a + 56

3d = 56

d = 56/3

So

( a ) + (a + 56/3)  + (a + 2*56/3)  + (a + 3*56/3)   =  360

4a + (6)(56/3)  = 360

4a +  112  =  360

4a  =  360  - 112

4a  =  248

a =  248  / 4    =   62  (degrees)

Let the center be O    and let the midpoint of UV =  P

Draw OP   and OV

Triangle  OPV is a right triangle  with angle VPO =  90

Cal the distance from O to UV =  d   and call the distance from OV  = r

And PV =  2

So

OV^2  = OP^2 + PV^2

r^2  = d^2 + 4            (1)

Likewise, we can let M be the midpoint of YZ

Connect  OM and OZ

OM = d + 2

OZ =  r

And MZ = 1

So   another right triangle is  formed such that

OZ^2 = OM^2 + MZ^2

r^2  = (d + 2)^2 +  1      (2)

Equate (1) and (2)

d^2 + 4 =  (d + 2)^2 + 1

d^2 + 4 = d^2 + 4d + 4  + 1

4d = -1

d = -1/4

Impossible....so....no solution to this