CPhill

Triangle HQN similar to Triangle FQG

HN / FG =  2 / 7

Therefore  HQ / FQ  = 2 / 7

Therefore  HQ / (HQ + FQ) =  HQ / HF  =  2 / ( 2 + 7)  = 2/9  ⇒ HQ = (2/9) HF

Triangle PFM similar to Triangle PHG

FM / HG =  3 / 8

Therefore PF / PH = 3 / 8

Therfore PF / (PF + PH)  = PF / HF  = 3 /( 3 + 8) = 3 / 11 ⇒  PF = (3/11) HF 

Getting a common denominator  between  9  and 11

HQ = 22/99FH

PF  = 27/99 FH

Therefore

FH - HQ - PF  =  PQ

FH - (22/99)FH - (27/99)FH  = PQ

(50/99)FH  = PQ

Therefore

PQ / FH  =  50 / 99

1/a  +   1/b =    (a + b)   /  (ab)

x^2 - 22x  - 9  =   0 

Sum of roots  = a + b =  - (-22) / 1  = 22

Product of roots  = ab  =  -9/1 =  -9

So

(a + b) / (ab) =   22 /  -9

(x  - 5) ( x + 5)  =  0 

x^2   - 25  =  0

x^2  +  0x  +   -  25  =   0

Rearrange as

x^2  + 26  =  0

x^2  =  -26              take  both roots

u =  i sqrt (26)              and      v  =  -i sqrt (26)

u / v  +   v /  u    =

i sqrt (26) / -i sqrt (26)    +     -i sqrt (26)  / i sqrt (26)  = 

- 1        +      -1    =

-2

https://web2.0calc.com/questions/rolls_2

Sorry....see my corrected answer

True....but i'm assuming that x and y are different integers.....

However....if  let them  be the same, then your answer is  correct

2. 

Let ABCDEFGH be right rectangular prism. The total surface area of the prism 30. Also, the sum of all the edges of the prism is 44. Find the length of the diagonal joining one corner of the prism to the opposite corner.

h = height

w = width

l = length

4 ( h + w + l)  =  44

h + w +  l =  11                   (1)

2 (lw + lh + hw) = 30          (2)

Square both sides of (1)

h^2 + 2 h l + 2 h w + l^2 + 2 l w + w^2  = 121       rearrange as

h^2 + l^2 + w^2 +  2 (lw + lh + hw)  = 121  (3)

Sub (2)  into (3)

h^2 + l^2 + w^2  +  30 =  121

h^2 + l^2 + w^2  = 91

sqrt  [ h^2 + l^2 + w^2 ] = sqrt (91)  ≈  9.54  =   length of the diagonal joining one corner of the prism to the opposite corner

CORRECTED

Set the equations equal

3  = 4x^2 + x  - 1         simplify

4x^2 + x - 4  = 0        divide through by 4

x^2  - x/4  - 1  = 0

x^2 + x/4  =   1                complete the square on x

x^2  + x/4  + 1/64  =  1 + 1/64

(x + 1/8)^2  = 65/64            take both roots

x + 1/8  =  sqrt(65) /  8                       x +  1/8  = -sqrt (65) / 8

x =   [ sqrt (65) - 1 ]  / 8                  x =  [-sqrt (65) - 1 ]  / 8

A =  ( [-sqrt (65) - 1 ] / 8 , 3 )

B = ([ sqrt (65) - 1 ] / 8 , 3 )

The distance between  these points =  2sqrt (65) / 8  =   sqrt (65)  /  4

m - n =  65  - 4  =   61

3a + 2b =   2       (1)     

a + 2b + 8a + mb  = k   →   9a + (2 + m)b   =  k    (2)

The system has infinite solutios when the equations are exactly the same

Multiply (1)  by 3

9a + 6b  =  6       (3)

Equating terms in (2) and (3)

(2 + m)  = 6           k =  6    

m  =  4