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I used x instead of z.....same difference

If 2 + 3i is a root, then so is  2 - 3i

So

( x - (2 + 3i) )  ( x - (2- 3i)   will divide  the polyomial.....simplify.....

x^2 - (2 + 3i)x - (2 - 3i)x  + (2 + 3i) (2 - 3i)

x^2  -2x -3ix - 2x + 3ix + (4 + 9)

x^2 -4x + 13

Using synthetic division

                          x^2    - x   - 2

x^2 - 4x + 13  [  x^4 - 5x^3 +15x^2 -5x - 26 ]

                          x^4 -4x^3 +13x^2

                          _________________________

                                   -x^3 + 2x^2 - 5x  - 26

                                   -x^3  +4x^2 -13x

                                 ___________________

                                              -2x^2  + 8x - 26

                                              -2x^2  + 8x - 26

                                            _______________

The remaining polynomial  is   x^2 - x - 2    which factors as  ( x + 1) (x - 2)

Seting each linear factor to 0  and solving for  x  produces the other two roots  x = - 1   and x = 2

y = 18x^2 - 4x + 20

y - 20  = 18 [ x^2 - 2/9 x  +1/81 - 1/81 ]

y - 20  =  18 ( x - 1/9)^2   - 2/9

y - 180/9 + 2/9  =  18 (x - 1/9)^2

(y - 178/9)  = 18(x - 1/9)^2

(1/18)(y - 178/9) = ( x - 1/9)^2

The vertex is  (1/9, 178/9)

4a = 1/18

a = 1/72

The focus is  ( 1/9, 178/9 + 1/72)  =   (1/9, 475/24)

N=20; R=0.07;PV=1000; P =0; FV=PV*(1 + R)^N
FV =$ 3,869.68

Circle

The center is (1,4)

The radius is 8

So...we have

(x-1)^2 + (y - 4)^2  = 8^2   implify

x^2 - 2x + 1 + y^2 - 8y + 16 = 64

x^2 + y^2 -2x - 8y  + 17 = 64   subtract 64 from both sides

x^2 + y^2 - 2x - 8y - 47 = 0

Good job, Melody  !!!!!!

Don't over-think this , Nickolas!!!

Your  first answer  of "the second one" was correct......!!!!

Let  a  = 3

Let b  = ( 4a - 2)

So

3*(4a - 2) = (4a - 2)*3

a*b  = b*a

32.

The original graph consists of the points $(x_1,f(x_1)),$ $(x_2,f(x_2)),$ and $(x_3,f(x_3))$. The graph of $y=2f(2x)$ consists of the points $\left(\frac{x_1}2,2f(x_1)\right),$ $\left(\frac{x_2}2,2f(x_2)\right),$ and $\left(\frac{x_3}2,2f(x_3)\right)$. Relative to the original graph, it is stretched vertically by a factor of $2$, but also compressed horizontally by the same factor. The vertical transformation doubles the area of the triangle formed by the three points, but the horizontal transformation halves it again, so the final area is equal to the original $\boxed{32}$.

Option A        ( 1, 2 , 5 , 5)

We have  these possible outcomes on two spins

(1, 1) ( 1,2) (2,1) (2,2) (1,5) (5,1) (2,5) (5,2) (5,5)

We have 3 outcomes whose sums are > 6  

So....the expected value is

$8(3/9) - $2(6/9) =

$ ( 8/3 - 4/3) =

$ (4/3) ≈  $1.33

Option B

We have 2^3  = 8 possible outcomes

Only one of these is 3 heads

So...the expected value is

$50(1/8) - $1(7/8) =

$ ( 50 - 7 ) / 8 =

$(43)/8 ≈ $5.38

Option B   is much better by   ≈  $ ( 5.38 - 1.33)  ≈ $ 4.05

For x^1/x = 25, there are only "complex solutions". However, for x^x = 25, there is a "real" solution, which is:

x = 2.96321977489........etc. So that:

[2.96321977489^2.96321977489] =~25

6, trust me

Your answer is just as valid as the book's

You get

(1/8)(x - 4) ( x^2 + 4x + 16)   which is actually the same as

(1/2)(1/4)(x - 4) ( x^2 + 4x + 16)  =

(1/2)(x - 4) (1/4) ( x^2 + 4x + 16)  =

(1/2 x - 2) (x^2/4 + x + 4) ⇒  "book answer"

Yours is actually easy to work with .....it avoids the nasty fractions in the terms

a = 1/2x    b = 2

(a-b)(a^2 + ab + b^2) =

(1/2x -2)(1/4 x^2 + x + 4)

Difference between successive terms = +4

27th term =   a1 + (n-1)(d)

                     -15 + (26)(4) = 89

Sn = n (a₁+a₂₇)/2 = 27 (-15+89)/2 = 999

Your answer is right :)

4P1 = 4 one-letter words.

4P2 =12 two-letter words.

4P3=24 three-letter words.

4P4 =4! =24 four-letter words.

Total =4 + 12 + 24 + 24 = 64 words.

(1/2x)^3 - 2^3 )

answer: (2x-2)(4x^2 + 4x+4) Correct?

Wow!!!!....that's impressive, Heureka!!!

Definitely one for my "Watchlist"

Yes, that is the link ! Read the solutions carefully, guest. 

Thank you so much got it!!

2x - 7 < 21    add 7 to both sides

2x < 28       divide both sides by 2

x < 14

So....this would be the sum of the integers 1 to 13, inclusive  =   13 * 14 / 2  =   91