CPhill

(3 + 2i) ( 4 + i)  =

4(3 + 2i) + i  ( 3 + 2i)  =

12 + 8i + 3i + 2i^2   =

12 + 11i + 2(-1) 

10 + 11i

(5 -  2i)^3  =

(5 - 2i) ( 5 - 2i) ( 5 - 2i) =

(25 - 5*2i - 5*2i + 4i^2) ( 5 - 2i) =

(25 - 20i + 4(-1) ) ( 5 - 2i) =

(21 - 20i) ( 5 - 2i) =

(105  - 100i - 42i + 40i^2 ) =

(105 - 142 i + 40(-1) ) =

 65 - 142 i 

(a + bi) ( a - bi)  =

a^2  + abi - abi - b^2i^2 =

a^2 - b^2(-1) =

a^2 + b^2

We want to know   when t  produces a min height

Note that a t = 0, the height = 100 ft

So....the t that produces a min height is given by   - (-60) / (2 * 10)  =  60 / 20  =   3 seconds

One method

If x + 4 is a factor then - 4 is a root

So

f(-4) should = 0

So we have    4(-4)^4 - 2(-4)^3  -52(-4)^2  - 15 (-4) + 8 =

1024  + 128 - 832 + 60 + 8  =  388...so .... not a root

Another method....synthetic division

-4    [  4     -2     - 52     - 15     8   ]

                -16      72      -80  380

       ________________________

          4     -18    20      -95   388            same remainder as above  !!!

If it was a root....we sould get 0  as a result

Assuming that t = 0   = 12 AM....the correct function is

h(t) = 0.6cos ( pi * t /6 ) + 1.6

c is correct

3pi * 4  =

12 pi

pi is irrational.....and a real non-zero multiple of an irrational is still an irrational

BTW.... b =  .0444... =   4/90 = 2/45....so 4 * 2 /45 = 8 /45  which is rational

MathWizard likes to play a fun number trick on her friends. She tells them to think of a number. She then tells them to subtract their number from 2 and multiply the result by 5. To this product she tells them to add half the difference when 24 is subtracted from 12 times their original number.

(a) If I use the number 8 and follow MathWizard's steps, what number will I get?

Mmmmmm....that's a lot of stuff..LOL!!!

2 - 8  = - 6

-6 * 5 = -30

"  half the difference when 24 is subtracted from 12 times their original number."   translates to

(1/2) ( 8 * 12 - 24 ]  =    (1/2) [ 96 - 24 ]  = (1/2)(72) = 36

So

-30 + 36 =

6 

The sum of the roots = -b / a =  -21/18  =  -7/6

The product of the roots = c / a = -400/18 = -200/9

Let the roots be  p, q

p + q =   -7/6    

pq = -200/9    ⇒   2pq = -400/9

(p + q)^2 =   49/36

p^2 + 2pq + q^2 = 49/36

p^2 +  (-400/9) + q^2  = 49/36

p^2 + q^2 =  49/36 + 400/9

p^2 + q^2 =  [ 49 + 1600] / 36  = 1649 / 36   ≈   45.8....just as EP found !!!!

(d)   y = 2 ( x + 1)^2 - 5             add 5 to both sides

y + 5 =  2 ( x + 1)^2             multiply both sides by 1/2

(1/2) ( y + 5) = ( x + 1)^2

The vertex is ( -1, -5)       and this opens upward

4p = 1/2

p = 1/8

The focus is ( -1, -5 + 1/8) = (-1, -39/8 )

The directrix is y = -5 - p =  - 5 - 1/8 =   -41/8  ⇒  y = -41/8

(c)   y = (1/16) ( x - 3)^2       easier!!!

16 y = (x - 3)^2

16 ( y - 0) = (x - 3)^2

The vertex is (3, 0)       and this turns upward

4p = 16

p = 4

The focus is  ( 3, 0 + p)  = ( 3, 0 + 4) = ( 3, 4)

The directrix is    y = 0 - p  = 0 - 4  =  -4  ⇒    y = -4

(b)   y = (-1/4) ( x + 2)^2 + 1

y - 1  =  -(1/4)(x + 2)^2

4 ( y - 1) = - ( x + 2)^2        the parabola turns downward

The vertex is ( -2, 1)

4p = 4      so   p = 1

The focus is ( -2, 1 - p) = (-2, 1 - 1 ) = (-2, 0)

The directrix is    y = 1 + p  =  1 + 1   = 2   ⇒    y = 2