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good job, history  !!!!!

18 + 5x^2 = 20x + 20

Rearrange as

5x^2 - 20x - 2  =  0      divide through by 2

x^2 -10x - 1  =  0

x^2 -10x  =  1             complete the  square on  x

x^2 - 10x + 25 =  1 + 25

(x - 5)^2  =  26

x -5 =   sqrt (26)   and -sqrt (26)

x = sqrt (26) + 5      →  rounded =  10

x= -sqrt (26) + 5   →   rounded  =  0 

Product =   10 * 0  =  0

Yooooo I finished quicker then CPhill. 

5x^2-20x-2=0. Using the quadratic formula, we get (20+-sqrt(440))/10. We can turn this into (20+-2sqrt(110))/10, or (10+-sqrt(110))/5. I calculated both answers, and got 4 and 0. 4*0 = 0, so I think the answer is 0. 

A = 2^(1/2) * 4^(1/6) =  2^(1/2)*(2^2)^(1/6)  = 2^(1/2)* 2^(2/6) = 2^(3/6)*2^(2/6) = 2^(5/6) = 2^(10/12)

B = (128)^(1/12) / 64^(1/3)  = (2^7)^(1/12) / (2^6)^(1/3) = 2^(7/12) / 2^(2) = 2^(7/12) / 2^(24/12) = 2^(-17/12)

C = (8^(1/5))^(2^2)  = 8^(1/5)^4 =  (2^3)^(4/5) = 2^(12/5) 

D = sqrt (4/ (2*8*4)) = sqrt (4/64) = sqrt (1/16)  = (1/16)^(1/2) = 1/4 = 1/2^2 = 2^(-2) = 2^(-24/12)

E= (2^(1/4)*4^(-1/2))^3  =  (2^(1/4) *(2^(2))^(-1/2))^3 = (2^(1/4)*2^(-1))^3 = (2^(-3/4))^3 = 2^(-9/4) = 2^(-27/12)

E , D, B , A , C

A = 2^(1/2)*4^(1/6) =1.781797436

B = (128)^(1/12)/64^(1/3)=0.3745767692

C = (8^(1/5))^(2^2)=5.278031643

D = sqrt(4/(2*8*4))=1 / 4

E = (2^(1/4)*4^(-1/2))^3=0.2102241038

Sorted from smallest to largest:
(E=0.2102241038,  D=0.25,  B=0.3745767692,  A=1.781797436,  C=5.278031643)

Oh... I thought he meant the absolute value. 

l c l  =   (-3)(4) - (5)(-2)  =    -12 - (- 10 )  =  -12 + 10 =   -2

Angle Bisector Theorem

BD /AB = DC/ AC            let BD = x ......so DC = BC - x =  5 - x

x/ 5 = (5-x) / 6

6x = 5(5-x)

6x = 25 - 5x

11x = 25

x = 25/11  = BD

Law of Cosines

AC^2  = AB^2 + BC^2 - 2(AB * BC) cos ABC

6^2 = 5^2 + 5^2 - 2(5 * 5)cos ABC

[36 - 50 ] / [ -50]  = cos ABC

cosABC =  -14/-50  = 7/25

Law of Cosines again

AD^2 =  AB^2 + BD^2 - 2(AB * BC) cos ABC

AD^2 = 5^2 + (5/11)^2 - 2 ( 5 * 5/11) (7/25)

AD^2 = 25 + 25/121 - (50/11)(7/25)

AD^2 = 2896 / 121

Er History I'm not sure...

I'm assuming you want the determinant of this matrix, which is \((-3)(4) + (-2)(5) = -22\). 

As small as x can  be is   -5

So....the bottom of the  range =  sqrt ( -5 + 5)  + sqrt (20 + -5) =  sqrt (15)

And  the function continues to  infinity no matter how  large the value of x

Range  =  [ sqrt (15) , inf)

Thank you so much :)

I suppose it is [3 5]
                        2 4

1234

2345

3456 

4567

5678

6789

789 10

8 9 10 11

.

.

.

16 17 18 19              looks like 16 sets to me

17 18 19 20 =116280  ====  too big

According to Angle Bisector Theorem:

CA / CB =AX / BX

2 / 4 =AX / 2

AX =[2 * 2] / 4 = 1

See the graph here :

The domain is  [ 0, 4 ]

(n)(n + 1)(n + 2)(n + 3) =n^4 + 6 n^3 + 11 n^2 + 6 n

1 = 24
2 = 120
3 = 360
4 = 840
5 = 1680
6 = 3024
7 = 5040
8 = 7920
9 = 11880
10 = 17160
11 = 24024
12 = 32760
13 = 43680
14 = 57120
15 = 73440
16 = 93024
 

Thank you TheOneXWZ, but I believe that there might be a mistake in one of your calculations. Thank you for helping, though!

Not sure what the question is....but maybe you want the hive to flower distance?

20 - 16 = 4 minutes of flying time .

distance / rate = time     or     rate * time = distance 

                                                9 *x  = distance              where 'x' = time at 9 ft/s

                                                6 ( 4-x)  = distance                these are equal 

                                                     9x = 6 (4-x) 

                                                      9x = 24 - 6x 

                                                       15x = 24 

                                                         x = 1.6 minutes at  9 ft/s           then distance = 9 ft/s * 1.6 min * 60 s / min = 864 feet