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There is   4 P 4 ways to arange the block of 4 math books ....and 4 ways to place this block int groupof 7 

     4 P4   * 4 = 24 * 4   = 96 ways to arrange and place the math books 

then there are   4 P 3 ways to arrange the remaining  Psych books = 24 ways 

24 x 96 = 2304   ways to arrange the books . 

First compute the denom  ....I will ASSUME both of the right terms are in the denominator thusly: 

(1+i) / ( (2-i)(3+i) ) 

(2-i)(3+i) = 

6 + 2i -3i - i^2 

6 - i +1

7 - i 

now you have 

(1+i) / (7-i)             multiply by   ( 7+i)  / (7+i)  to get 

(6+8i)/ 50     which reduces to    (3+4i)/25

Thanx, Chris !    

~EP

.....as an edit...the question asked for radicals in the answer 

                 x = 2 + 2 sqrt(5)     y = 6 + 2 sqrt(5) 

First,  simplify/reduce it to :

2x ^2 -12x + 18 =0      factor out '2'

2 ( x^2 -6x + 9) = 0       now it ieasier to factor 

2 ( x -3)(x-3)  = 0             shows x = 3 

Let's assume there are 100 games :

.75 * 100 * .35  = games won so far 

.25 (100) x       games remaining  times the percentage needed to get to .5(100)  <==== wins = losses

.75 (100) .35   +   .25 (100) (x)  = .5 (100)

     26.25         +  25 x = 50 

          25 x = 23.75

             x = .95                     You need to win 95% of the remaining games                

Nice, EP   !!!!

Volumes =  

Volume of outside cylinder - Volume of  inside cylinder  

First solid  volume

pi (3 - -2)^2 * 5  -  pi(0 - - 2)^2 * 5 =

pi [ 125  - 20]  =

105 pi

Second  solid volume

pi  (5 - -5)^2 * 3  -  pi (0 - -5)^2 * 3  =

pi [ 300 - 75]  =

225 pi

Difference in  volumes  =    (225 - 105 )  pi   =   120 pi

Let P  = (x , 6-x)

We can equate distances (squared)

(10 - x)^2  + ( -12 - (6-x)^2   =  ( 2 -x)^2  + (8 - (6-x) )^2

(x -10)^2  + ( x - 18)^2  = (x - 2)^2 +  ( x + 2)^2

x^2 -20x + 100  + x^2 -36x + 324  =  x^2 - 4x + 4  + x^2 + 4x + 4

2x^2  - 56x + 424 =  2x^2 + 8

416  =  56x

x = 416 / 56   =   52  /  7

y = 6 - (52/7)  =  -10/7

(x, y)  = (52/7 , -10/7)

Find max of   x+y    start with standard circle form 

x^2 -4x    + y^2 - 12y   = 0 

(x-2)^2    + (y-6)^2   =  4 + 36 

(x-2)^2   + (y-6)^2 =  40

Now find the 45 degree line that goes through the center   (2,6) 

             slope    = 1          y = x + b    shows b = 4 

                   so the line is y = x + 4 

                     where the circle and the line intersect will be the max   x+y 

x+4 = y 

sub into the circle equation and solve for x = 6.47124      then y = 10.47124

Mx value of  (x+y)   is 16.94428

Here is a graph: 

[ABCD]  =  [ EFGH] - [EBA] - [ FCB] - [CGD] - [ ADH]

[ABCD ] = (5 *3) - (1/2) [ 1 *2  + 4*1  + 2* (1/2)  + 4.5 *1 ]  =

15  - (1/2)  [ 2 + 4 + 1 + 4.5 ]  =

15  - (1/2) [ 11.5 ]  =

15 - 11.5 / 2  =

[30 - 11.5 ]  / 2 =   

9.25

(3 , 19)

(19, 3)

(5, 17)

(17, 5)

Pretty easy

(b^2 - a^2) / ( b -a) =  2

(b + a) (b - a) / (b - a)  =  2

b + a  =  2

If I could figure this  out, I'd be qualified to fly the Space Shuttle.....

Yikes !   

The radius is shown as '2'....I think...    the diameter = 2 * radius = 4  units 

For an absolute answer , I think I need more info.....

\(\lfloor t\rfloor = 3t + 4 - \lfloor 2t \rfloor \)

(blue function on graph)  = (green function on graph)

Interesting !!!! .....the functions  never intersect.....so.....no solutions  for t

60pi = pi * r^2 * (83/360)

60  / (83/360)  = r^2

r = sqrt  [ 360 * 60 / 83 ] ≈  16.13 

See image below :

(2x + 7) (x - 4) =  -31 + jx

2x^2 - x - 28  = -31  + jx

2x^2 - (1 + j) + 3 =  0

To have only one solution

(1 + j)^2 - 4(2)(3)  = 0

(1 + j)^2  - 24  =  0 

(1 + j)^2  =  24           take both roots

j + 1 =  sqrt (24)              j + 1 = -sqrt (24)

j  + 1 = 2sqrt (6)             j + 1 = -2sqrt (6)

j = -1 + 2sqrt (6)             j =  -1 - 2sqrt (6)   

Using the vertical line test.....if a vertical line passes through the graph only ONE time ( no matter where the vertical line is placed)  the graph is a function 

a) no

b) no

c) no 

d) yes      this is the graph of    f(x) = | x - d |    where 'd' is the distance from the origin to the vertex