CPhill

Let b = (-1/2)

So

x^4 - 3x^2 + 2  + (-1/2) [ 2x^4 - 6x^2 + 2x - 1 ]  =

x^4  - 3x^2 + 2   - x^4 + 3x^2 - x + 1   =

-x + 3

The smallest possible degree ( as found by the guest)  =   1

\( $\lfloor \sqrt[3]{n} \rfloor = 5$\)

Note that     (125)^(1/3)  =  5

And that      (216)^(1/3)  = 6

So....the number of positive integers that satisfy this  =     215 - 125 + 1 =  91

        B

A       8      D      3     C

Note that  triangles  ABD  and DBC are under the same height

And....triangles under the same height are to each other as their bases

So

[ABD ]   =  [ 8 / ( 8 + 3)  ] [ ABC]   =  (8/11) (120)  =    960 / 11 

cos (x)  = .35

arccos (.35)  = x ≈   69.5°

OK

We have that

-6t^2 + 51t    =   108         divide through by -3

2t^2  - 17t  =  -36

2t^2 - 17t + 36 =  0      see if we  can  factor   (you could use the Q Formula but I like to  factor, if possible )

(2t - 9) ( t - 4)  =  0

2t - 9  = 0                                 or            t  - 4   =  0

2t = 9                                                       t = 4 sec

t = 9/2 sec = 4.5 sec

It first reaches 108 ft  at 4 sec

Welcome....!!!

BTW -  no equation shown for the second one

(x - 2) ( 2x + 5)  = 8x - 6

2x^2 - 4x + 5x - 10 = 8x -6

2x^2 + x -10 =  8x - 6       rearrange as

2x^2 - 7x - 4  = 0

(2x + 1) ( x - 4)  =  0

The second factor set = 0  is what we want 

x - 4 =  0 

x  = 4

3ax + 3y   =  5b

2x + y = 5   multiply through by 3  →  6x + 3y  = 15

To have infinite solutions the equations must be the same ......this implies that

3a =  6         and           5b  = 15  

a  =   2                              b =  3

No graph shown  ....

Rearrange as

sqrt (10x -17)  = sqrt (5x - 102)     square both sides

10x  -  17 =   5x  -102

10x  -5x  =  -102 + 17

5x  =  -85

x = -85/ 5  =  -17

Note that

10 (-17) - 17 = - 187

And

5(-17) - 102 =  -187

So  x = -17  is the  solution (assuming that complex forms  are OK )  and we actually have

sqrt ( -187) - sqrt (-187)  = 0

sqrt (187) i  -  sqrt (187) i  =  0

 i ( sqrt (187) - sqrt (187) )  = 0

i (0)  = 0   

0i = 0   

0 = 0   is true