CPhill

Solve this for the x coordinate of the vertex  and axis of symmetry

6x - 2  =  0

Add 2 to both sides

6x  = 2

Divide both sides by 6

x  =  1/3

And the y coordinate  is  - 7

So......

(1/3, -7) , x  = 1/3   shifted to the right 1/3 units and down 7 units 

The last one

y  =  -16x^2 + 105x  + 12

Time to reach max ht  =    -105 / [ 2 * -16]  =  105/32  sec

Put  this into the function to find the max height  =

-16 (105/32)^2 + 105(105/32)  + 12   =  184.27 ft

And the time it takes to hit the ground  =

-16x^2  + 105x  + 12  =  0

( -105 ±√ [ 105^2   - 4(-16)(12) ]  )  /  [ 2 (-16)] 

( -105 ±√ 11793 )  /  -32     take the negative sign

( -105 - √11793 )  / -32  ≈  6.67  sec

We  have this  :

350 / tan (2°)  -  350 / tan (5.5°)  =  distance   =  6387.8 ft

x^4  + 5x^3  + 7x^2  - 3x - 10  = 0

If we  add the coefficients and constant term and get  0 then 1 is a root....so....

1 + 5 + 7  - 3  - 10   =    13  - 13   =    0

So 1  is a root

So.....we need to check  wheter 2  is a root...if not....then  -2 must be the other  rational root

So

(2)^4  + 5(4)^3 + 7(2)^2  - 3(2) - 10   >  0

So   -2  is the other rational root

So    -2, 1   are the rational roots

32pi  =  2*pi *r* (240/ 360)        divide both sides by pi

32  =   2 * (240/360) * r

32  = 240/180 * r

32  =  (4/3) * r

(32) (3/4)   = r

24  =  r

36pi  =  2* pi * 48  *  ( Angle / 360)

36    = (96/360)  * Angle

36 (360 / 96)  = Angle

(36/96)(360)  = Angle

(3/8)(360)  - Angle

135°  = Angle  =   (3/4)pi   rads

Quadrant iV ⇒  cosine positive, sine negative

4pi/5  *  (180) / pi  =  (4/5)(180)  =  144°

posive coterminal angle  = 2*360 - 32  =  688°

negative coterminal angle  = -360 - 32  =  -392°

[ if the negative coterminal angle is supposed to be between - 500 and -1000, we have 

-2*360 - 32  =  -752°  ]

257°   lies between  180°  and 270°....so......Quadrant III

3x^2 - 4 =  -x^4         add x^4 to both sides

x^4 + 3x^2  - 4  = 0     factor as

(x^2 + 4) (x^2 - 1)  = 0

(x^2 + 4)  (x + 1)  ( x-1)  = 0

Setting the last two factors to 0 and solving for x produces   x = -1 and x  = 1

Settting the first factor to 0, we have

x^2 +4  = 0   subtract 4 from both sides

x^2  =   -4         take both roots

x = ±√-4    =  ±2i

So the solutions are    1, -1, 2i, 2i

(s + 4v)^5  =

s^5  +  5(s^4)(4v) + 10(s^3)(4v)^2  + 10(s^2)(4v)^3  + 5(s)(4v)^4  + (4v)^5  

s^5  +  20s^4v  + 160s^3v^2   + 640s^2v^3  + 1280sv^4  +  1024v^5  [third answer ]

Ah!!!....that's more elegant than mine, for sure  !!!