CPhill

Simplify as

x^2 - 6x + 13

uv  =  13

2uv  = 26

u + v =  6       square both sides

u^2 + 2uv + v^2  = 36

u^2 + 26 + v^2  = 36

u^2 + v^2  = 10

u/v^2  + v/u^2  =  [ u^3 + v^3 ] / [ uv]^2

u^3 + v^3  = ( u + v) ( u^2 + v^2 - uv)  = (6)(10 - 13)  =  -18

[uv]^2 = [ 13]^2   = 169

u/v^2  + v/u^2  =  [ u^3 + v^3 ] / [ uv]^2 =  -18 / 169

Simplify as

10x^2 + 10x - 1

ab = -1/10

2ab = -2/10  = -1/5

a + b  =  -1      square both sides

a^2 + 2ab + b^2  = 1

a^2 - 1/5 + b^2  = 1

a^2 + b^2  = 1 + 1/5  =  6/5

(a -4) / ( b -4)  +  ( b -4) / (a -4)  =     [  (a - 4)^2  + ( b -4)^2] ]  ' [ (a -4) ( b-4) ]  =

[ a^2 - 8a +16  + b^2 - 8b + 16 ]  /  [ ab - 4(a + b) +16 ]   =

[  (a^2 + b^2) - 8(a + b) + 32 ]  / [ ab - 4 (a + b) + 16 ]  =

[ 6/5 - 8 (-1) + 32 ]  [ -1/10 - 4 (-1) + 16 ]  = 

[ 6/5 + 40 ]  / [ 20 - 1/10] =   20497 / 25

A²  = [1/sqrt 2    -1/sqrt 2]    *   [  1/sqrt 2   -1/sqrt2 ]   =   [ 0  -1 ]

         [ 1/sqrt 2    1/sqrt 2]         [   1/sqrt 2   1/sqrt2]         [  1  0 ]

A⁴ = A² * A²  =  [ 0 -1 ]  *  [ 0 -1 ]  =    [ -1  0 ]

                          [ 1 0 ]      [ 1 0  ]        [  0  -1]

A⁶ =  A⁴ * A²  =  [ -1 0 ]  * [ 0 -1]  =  [ 0  1 ]

                          [ 0  -1 ]    [ 1 0 ]      [ -1 0]

A¹² = (A⁶)²  =   [ 0    1 ]

                           [ --1  0 ]

Note that   (A⁶)ⁿ  =  [ 0 1  ]          where n = 1,2,3,4.......

                                [ -1  0]

A¹⁰² = (A⁶)¹⁷  = [ 0 1 ]

                          [-1 0 ] 

B²  = [ 1 1  ]  *   [ 1 1  ]  = [ 1  2 ]

         [ 0   1 ]     [ 0  1]      [ 0  1]

B⁴  = [ 1 2 ]  * [ 1  2 ]  =   [1  4 ]

         [ 0  1]     [0  1 ]       [ 0  1]

B⁶  = B⁴ * B²  = [ 1 4 ]  *  [ 1 2 ]  =  [  1  6 ]

                         [ 0  1]      [ 0 1]       [   0  1]

Note that  B²ⁿ = [ 1  2n ]     where n =1,2,3,4.......

                          [ 0   1]

B¹⁰² = B^2*51  = [ 1  2*51 ]  =  [  1   102 ]

                           [ 0    1  ]       [  0     1  ]

A¹⁰²B¹⁰²   =  [  0  1 ]  *  [ 1   102 ]  =    [ 0       1 ]

                      [ -1  0 ]    [  0     1  ]         [  -1 -102 ]

B¹⁰²  A¹⁰² =  [ 1  102 ] *   [ 0 1  ]  =  [ -102   1 ] 

                     [ 0       1]      [ -1 0 ]      [  -1      0 ]

A¹⁰²B¹⁰² - B¹⁰²A¹⁰²  =  [  102     0  ]

                                       [   0    -102 ] 

See the following diagram

The slackrope walker is at "E"

We can form  two right triangles

One has a legs of (15 -3)   =  12   and 5

Then the hypotenuse  od this triangle =  sqrt [12^2 + 5^2] = sqrt (169) = 13  =  (DE)

This is part of the rope length

The other right triangle has legs of 12  and (14 - 5) = 9

The hypotenuse of this triangle is  sqrt [ 12^2 + 9^2]  =  sqrt [225]  = 15 = (BE)

This is the other part of the rope

The total length of the rope is (DE) + (BE) =    13 + 15  ≈  28 ft 

I think this  might be the question :

In the figure, angles ABC and AST are right angles. If AC = 35, AT = 11, and BT = 9, then what is AS?

AB = AT + BT =  11 + 9  = 20

Triangle AST  is similar to triangle ABC

AS / AT  =  AB / AC

AS / 11  = 20 / 35

AS = (11 * 20 ) / 35  =   220 / 35  =   44 / 7

I don't know anything about modular math, but here's an answer from a "Guest."

This one is relatively simple. You have to take each modulus x some constant + remainder = each other as follows:

(6 * a) + r =(20 * b) + 9 =(45 * c ) + 4=x. You simply try, by iteration, to find the values of a, b, and c.

If you look at (45*c) + 4 and try c= 1, then it becomes: 45*1+4=49. and if you look at 20b+9, you will see that b= 2, and it becomes 20*2 + 9 =49. Check and see if it balances 6*a +r. If you take 49 and divide by 6, you will get 8, because 6*8 =48. And so 49 mod 6 =1. Therefore, r can only =1.

Since the LCM of 6, 20, 45 =180, therefore x will be:

x =180C + 49, and C =0, 1, 2, 3, 4, 5.......etc. So:

x =180*0 + 49 =49

x =180*1 + 49 =229

x =180*2 +49 =409

x =180*3+49 =589..........and so on. All these values of x satisfy the congruences.

See proyaop's answer here : https://web2.0calc.com/questions/square_34

https://web2.0calc.com/questions/algebra_10763

(n-3) + (n-2) + (n-1) + n + (n+1) + (n+2) + (n+ 3)  =  (28/9)(n + 3)

7n  = (28/9)(n + 3)

7n = (28/9)n + 28/3

63n = 28n + 84

63n - 28n = 84

35n = 84

n = 84 / 35  =  12 / 5

(n - 3)  = (12/5 - 3)  =  -3 / 5      (which is NOT an integer !!! )

Let A  =  [  1  0 ]             Let B  = [ a  b ]

              [  0  1 ]                           [ c  d ]

AB  =  [ a  b]

           [ c  d ]

BA  = [ a  b ]

          [ c  d ]