CPhill

a - b = -2 (c + d)    (1)

b = a - 5                 (2)

d = c + 3                 (3)

Sub (2) , (3) into (1)

a - (a -5)  = -2 (c + c + 3)

5 = -2 ( 2c + 3)

5 = -4c - 6

11 =  -4c

c = -11 / 4

(22 - 5)  * N  =

17 * N

We will only be considering those  who  did some homework = 500 - 40  = 460

n ( M U E U S)  =

n(M) + n(E) + n(S)  - n (M  ∩ E) - n(M  ∩ S) - n(E  ∩ S) + n (M  ∩ E  ∩ S)

460  =  259+ 219 + 238 - 85 - 81 - 108 + n (M  ∩ E  ∩ S)

460  =   442 +  n (M  ∩ E  ∩ S)

460 - 442  =   n (M  ∩ E  ∩ S)

18 =  n (M  ∩ E  ∩ S)

1 AU ≈  150,000,000 km

So

.72AU  =  .72 ( 150,000,000) km ≈ 108,000,000 km

n (A U B)  = n (A) + n(B)  - n ( A  ∩ B)

      9  =    2B + B     -   3

      12  = 3B

      12/3 = B =  4

      A  =  8

24551 is  prime....but 1 divides  every real number

           A

                 H

B                   C

AH = (2/3) (4) =  8/3

HC = 4/3

Triangle BAH  is right with BHA = 90

So

BH = sqrt (AB^2 - AH^2)  = sqrt ( 4^2 - (8/3)^2)  = sqrt [ 16 - 64/9 ]  = sqrt [ (144 - 64) ] / 3 = sqrt (80 ) / 3  

Triangle HBC is right with  BHC = 90

BC  =sqrt ( BH^2 + HC^2)  = sqrt ( (sqrt (80) / 3)^2  + (4/3)^2 )  = sqrt [ 80/9 + 16/9]  = sqrt ( 96) / 3 = 

4sqrt (6) /  3

AD = sqrt ( AE^2 + DE^2)  =sqrt (7^2 + 24^2)  = sqrt (625)  = 25 = BC

FC =  sqrt ( BC^2 - FB^2)  =sqrt (25^2  - 15^2)  =sqrt ( 625 -225)    = sqrt (400)  = 20

In triangle  BFC  draw altitude FG

Let GC = x    Let BG = 25 -x

FG^2 = FB^2 - BG^2 =  FB^2 - (25-x)^2  = 15^2 - (25 -x)^2

FG^2 =  FC^2  - CG^2 = FC^2 - x^2  =  20^2 - x ^2

So

FG^2 = FG^2

15^2 - (25-x)^2  =  20^2 - x^2

225 - 625 + 50x - x^2  = 400 - x^2

50x = 800

x = 16 = CG

So  FG =  sqrt (FC^2 - CG^2) = sqrt ( 20^2 - 16^2)  = sqrt (400 - 256) = sqrt (144)  = 12

Extend FG  through F  and  let it  intersect   AD  at  I

Triangle IDF is a right triangle with angle DIF = 90

Note that DI = CG

tan angle ADE  = AE /DE = 7/24

tan angle ADE = tan angle IDF  = FI /DI  = FI /16

So

FI /16  = 7/24

FI = (7/24) (16)  = (2/3) (7) =  14/3

AB = FG + FI  =  12 + 14/3 =   50 / 3

f(2) =  (2)^2  + (2)^3  =  4 + 8  = 12

Re-write as  ax^2 + 2x + 46  = 0 

If we only have one solution , the discriminant =  0

So

2^2 - 4(a) (46)  = 0

4 - 184a =  0

4 = 184a

a = 4/184  = 1 / 46