CPhill

Note that      [a + 1/b ] / [a -1/b ] =  [  (ab +1) / b ] /  [ (ab  -1) / b ]  =  (ab + 1) / (ab  -1)

Note that ab will be negative   and the  numerator and  denominator will always have an  absolute  value difference  of  2

The smallest possible  fraction results when   a = -2 and  b = 3  =   -5/- 7 = 5/7

The function  is  maximized when a =  -6  and  b  = 5....so....

(-6*5 + 1)  / (-6*5  - 1) =      -29   / -31   =  29 / 31

2x^2 + 5x + c =  x^2 + 7x    rearrange as

x^2 - 2x + c  = 0

This will have  at least one  real solution when  the discriminant ≥   0

So

(-2)^2  - 4(1)(c) ≥ 0

4 - 4c ≥  0

4 ≥   4c

1 ≥ c

So....it will have at least one real solution when c ≤ 1

x^2-25x+80 = -28x+75

x^2 + 3x + 5  = 0

Note that       a/b  +  b/a =   [a^2 + b^2 ] / [ab ]

Sum of roots = a + b =  -3     square both sides   

a^2 + 2ab + b^2  =  9        (1)

Product of roots = 5

So  ab = 5

2ab = 10       (2)

Sub (2) into (1)

a^2 +  10 + b^2    =  9

a^2 + b^2  =   -1

So  a/b + b/a =   -1 / 5

We have these two equations

1 = a (1 - 2)^2 + k   →    1 = a + k        (1)

-9 = a (4 - 2)^2 + k  →    -9 = 4a +k      (2)

Subtract (2) from (1)

10 = -3a

a = -10/3

k = 1 - a  = 1 - (-10/3) = 13/3

So the polynomial is

y = (-10/3)(x - 2)^2  + 13/3

And we  are trying to  find the  roots

So

(-10/3) ( x - 2)^2  + 13/3  = 0                     multiply through by 3

-10(x - 2)^2 + 13 =  0

(x - 2)^2  =  13/10            take the  positive root

x - 2 = sqrt (13/10)

x =  2 + sqrt (13/10)

Check your question.....your  root is not  correct

The number of problems he solves on  a  normal  day  =  p*t

On the day he drinks coffee  he solves 2pt problems....so...

(4p + 7) ( t - 3)  = 2pt

4pt +7t - 12p - 21  = 2pt

2pt +  7t - 12p - 21 =  0

2pt + 7t = 12p + 21

t =  [  12p + 21 ]  / [ 2p + 7 ] 

Note that  this makes t an nteger when   p= 7  and t =  5

So.....he solves    2(7)(5) =    70 problems on the  day  he  drinks  coffee

Call the area of the base =  B

Call the area of  each triangular face B / 2

So 

 B + 4(B / 2)  = 432

B + 2B =  432

3B = 432

B = 144

So ...the side of  the base = sqrt (144)  =  12

And the area  of  each triangular face  =   B / 2 =  72

The slant height of the  pyramid = 

area of triangular face  = (1/2) (slant height) (base side)

72 = (1/2)(slant height)(12)

144 = (slant height) (12)

12 = slant height

Height of pyramid =  sqrt [ slant height^2 - (base length / 2)^2 ] = sqrt [ 12^2 - 6^2] =

sqrt (108)  = sqrt (36 * 3)  = 6sqrt3

Volume of  pyramid = (1/3) (base area) (height)  = (1/3)(144)(6sqrt 3)  =  288sqrt 3  units^3

One equation , three unknowns....we may have multiple solutions !!!

WolframAlpha actually shows three different solutions :

https://www.wolframalpha.com/input?i=6xyz+%2B+30xy+%2B+3xz+%2B+8yz+%2B+15x+%2B+44y+%2B+8z+%3D1342

C (48 , 8)  =  377,348,994  different  choices

Convert inches to  feet :

108 / 12  =  9 ft

9 ft /  2 =   4.5  ft  = the length of each piece

3pi / 16