CPhill

The quantity inside the sqrt  must be ≥ 0

So

x^2 -8x + 5  ≥  0

Find the two roots of this quadratic

x  =  [ 8 +/- sqrt (64 -20] / 2

x = [ 8 + sqrt 44 ] /2       and   x  = [ 8 -sqrt 44] /2

Since x^2 -8x + 5  is a parabola that turns upward....the solution includes the roots (because of the ≥ sign) and also includes the intervals outside the roots  →

( -inf , (8 -sqrt 44)/2 ]  U  [ (8 + sqrt 44 ) / 2 , inf ) = the domain

THX, mercurym999 !!!

Another method using the Law of Cosines  (and assuming a radius of 1 )

d^2  =  1^2  + 1^2  - 2 ( 1 * 1)  cos (127° -37°)

d^2  = 2  - 2  cos (90°)

d^2  = 2 - 2 (0)

d^2  = 2

d =  sqrt 2

a)  increases on   (0, inf)

     decreases on  (-inf, 0)

b)  derivative =  2x

c)  sign on f'(x) from (-inf, 0)  =  -

     sign on f'(x) from (0, inf) = +

\( \sin 4A = \sin A + \sin 2A \)

sin ( 2(2A))  = sin A + sin 2A

2sin2Acos2A  = sinA + sin2A

2sin2A ( 1 - 2sin^2A)  = sinA + sin2A

2sin2A  - 4sin2A(sin^2A) = sinA + sin2A

sin(2A) - 4sin(2A)* (sin^2( A))  = sin(A)

2sinAcosA - 8sinAcosA (sin^2A) = sinA

sinA [ 2cosA - 8cosA (1 -cos^2A) ] - 1 =  0

sinA  = 0     A =  0”

2cosA - 8cosA (1 -cos^2 A) = 1

2cosA - 8cosA + 8cos^3A - 1 = 0

8cos^3 A - 6cosA - 1 =  0

Let  A  = x

8x^3 - 6x - 1   =  0

Using WolframAlpha to solve this  cubic   gives  x ≈  .93969

cos A  =.93969

arccos (.93969)  = A  =  20°

Thx, Bosco  !!!

Another way

first term  = a

second term  = a + d

third term  =  a + 2d

a + ( a + d) + (a + 2d)  = 9

3a + 3d  = 9

a + d =  3

a =  3 - d

a    d           series

0    3            0  3  6

1    2            1  3  5

2    1            2  3  4

3    0            3  3  3   ( if  "d"  allowed to  = 0 )

4   -1            4  3  2

5   -2            5  3  1

6   -3            6  3  0

Same answer as Bosco !!!!

y = 14  ( x^2 - x/14  - 1/14)       complete the square on  x

y  =  14 ( x^2 - x/14 + 1/784 - 1/14 - 1/784)

y = 14 ( x^2 - x/14 + 1/784  -  57/784)

y= 14 ( x - 1/28)^2  -  57/56

We have the form

y = a ( x - h)^2 + k

The focus  is found  at    (  h ,  k + 1/(4a) )  =  ( 1/28 , -57/56 + 1/56) =     ( 1/28 , -57/56 + 1/56)  =  ( 1/28 , -1)

28  = (1/2)  (  angle ROB  - angle SOT)

56  =  (3 angle SOT  - angle SOT)

56  = 2 angle SOT

28  = angle SOT

3 * angle SOT  = angle ROB =   84

minor arc RS    =   angle  ROS  =  180 - angle SOT - angle ROB =    180 - 28 - 56  =    68°

GOOD JOB !!!!!

https://web2.0calc.com/questions/geometry_80541

x^2 - 2x + y^2 - 4y = 0

x^2 - 2x + 1 + y^2  - 4y + 4  =  5

(x - 1)^2 + ( y - 2)^2  =  5

max  (x + y)  =   1  +  r cos 45 °  +   2  + r sin 45°  =   

1 + sqrt (5)  (1/sqrt 2)  +  2  +  sqrt (5) (1/sqrt 2)

3 +  2 sqrt (5) / sqrt (2)  =

3 +  sqrt (2) * sqrt (5)  =

3  + 2 sqrt (10) /2    

3 + sqrt (10)

x / r =  [ x + h ] / R

x (R / r ) = x + h

x (R/r) - x =  h

x [( R/r) - 1 ]   = h

x  =    h / [ (R/r) - 1 ]

H = x + h

H = h / [ (R/r) -1 ] + h

"Anchor" any one of the groups  and  arrange them in two ways

Choose any one of the  three remainig groups and arrange this  group in two ways

Choose any one of the two remaing groups and arrange this  group in two ways

Arrange the remaining group in two ways

2 * C(3,1) * 2 * C(2,1)* 2 * 2  =   96 ways