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(a) Let w = a + bi and z = c + di.  The rest is expanding.

(b) Let w = a + bi and z = c + di.  Then

\[\dfrac{w + z}{1 + wz} = \dfrac{a + c + bi + di}{1 + (a + bi)(c + di)}\]

To express this in rectangular form, we can multiply the numerator and denominator by the conjugate:

\[\dfrac{a + c + bi + di}{1 + (a + bi)(c + di)} = \dfrac{(a + c + bi + di)((1 - (a + bi)(c + di))}{(1 + (a + bi)(c + di))(1 - (a + bi)(c + di))}\]

The denominator simplifies to (1 - (a^2 + b^2)(c^2 + d^2)), which is real.  The numerator simplifies to a^2 - b^2 + c^2 - d^2, which is also real.  Therefore, the complex number (z + w)/(zw + 1) is real.

\(Solve for\ x:\\ \dfrac{a}{x-a} + \dfrac b{x-b} = \dfrac{2c}{x-c} \)

Hello Guest!

\(\dfrac{a(x-b)(x-c)+b(x-a)(x-c)}{(x-a)(x-b)(x-c)}=\dfrac{2c(x-a)(x-b)}{(x-a)(x-b)(x-c)}\)

\(a(x-b)(x-c)+b(x-a)(x-c)=2c(x-a)(x-b)\)

wolfram alpha  \(⇒ \)

\(2abc-2abx-acx+ax^2-bcx+bx^2=2abc-2acx-2bcx+2cx^2\) 

\(a+b-2c\neq0,\ x=\dfrac{2ab-ac-bc}{a+b+c}\)

\(b=a, c=a\)

\(x=0\)

  !

125% =  1.25

"Of" means "Multiply"

So

1.25n   =30       divide both sides  by 1.25

n = 30/1.25   =   24

25%   =   .25 =  25/100 =  1/4

So

(1/4)  of 24   =    24 (1/4)   =    6

He reaches a min height  in      5/(2*1) =  2.5 s

So    the min height is

(2.5)^2  - 5(2.5) + 29    =  22.75 ft  =   23 ft

radius of  circle =   31000  light years

Number of  m  in one light year  =    3 * 10^8 / s  *  (3600 * 24 * 365)s   = 9.4608 * 10^15   m

Converting this to km   (divide by 10^3)  =   9.4608 * 10^12  km   in one light year

So   .....radius of this  circle =   31,000 * 9.4608 * 10^12 km  = 2.932848 * 10^17 km

Total distance  traveled by planet in one orbit =  2pi * r =   2 pi * 293.2848 * 10^17km

Convert  286,000,000 years  to hrs =   286,000,000  * 24 * 365  = 2.50536 * 10^12  hrs 

So....the speed is

D / T  =         [ 2pi * 293.2848*10^17] km  /  [ 2.50536 * 10^12  ]hr    ≈ 7.36 * 10^7   km/hr

What is the  method of solving this  ???

The center of  the  x^2 + y^2 + 1  =  (0,0)

The  radius of  Gamma  =   ( a + 2)  =  r

The distance  from (16,a)  to (0, 0) =  (r + 1)  =   ( a + 2 + 1) = (a + 3) 

16^2  +  a^2   =  ( a+ 2 + 1)^2

256 + a^2  =  (a + 3)^2

256  + a^2  =   a^2  + 6a  +  9

6a  =  247

a = 247/6

See the graph here  : https://www.desmos.com/calculator/l9klwgdmus

Your numbers are wrong!!!

Thank you I get it!

im sorry for number 3 its The x-intercepts of the parabola y=x^2+bx are (-3,0)  and  (5,0) Find the equation of the parabola, and submit your answer in y=ax^2+bx+c form.

The   x coordinate  of vertex  will   occur   halfway  between the x intercepts  =  (5 - 3) /2  =  1

And the x coordinate of the  vertex =   -b / (2a)       where a   = 1   

So

-b / 2  =  1

b  =  -2

And   since (5,0)  is on the  graphm we can solve for  c  thusly

0 =  5^2  - 2(5)  + c

0 = 15 + c

c =  -15

So  the equation  is

y = x^2  -2x  - 15

In triangle ABC, D is the midpoint of BC. Given that AB=3cm, AC=5cm, and BC=7cm, find AD.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

AD = 1/2 {sqrt(2*3² + 2*5² - 7²)}

AD = 1/2 √19  

Here's the answer

2 3

5 3

                A

       b                c

C             a                B

AD  is a  median  drawn from A

Formula for  this median length   is

(1/2)  sqrt  [ 2 (b^2 + c^2)   - a^2 ]   =

(1/2)  sqrt   [ 2 ( AC^2 + AB^2)   -  BC^2  ]  =

(1/2) sqrt  [  2  ( 5^2  + 3^2)  -  7^2  ]  =

(1/2)  sqrt [ 2 ( 34 )  - 49  ]  =

(1/2) sqrt  [ 68 - 49  ]   =

(1/2)  sqrt (19)  =  

sqrt (19)/2

im sorry for number 3 its The x-intercepts of the parabola y=x^2+bx are (-3,0)  and  (5,0) Find the equation of the parabola, and submit your answer in y=ax^2+bx+c form.

(b) There are 108/3 = 36 unique sets.

(c) For the first card, there are 3 ways to choose the number.  For the second, there 2 choices, and then there is only 1 choice.  So there are 3*2*1 = 6 ways that the numbers can be chosen.  Doing this for the other attributes, we get 6*6*24*6 ways.  But the order of cards doesn't matter in a set, so we divide by 3!: 6*6*24*6/3! = 864.  So there are 864 sets for part (c).

(d) The cards can have the same number, color, shape, or shading.  If all the colors are the same, then there are 6*24*6/3! = 144 sets.  If all the numbers are the same, then there are 144 sets.  We get the same number for shape and shading, so there are 4*144 = 576 sets for part (d).

(e) We need to choose two of the attributes.  There are C(4,2) = 6 ways of choosing two attributes.  For each of these two attributes, there are 3 options.  For the other two attributes, there are 3 ways of assigning the choices, so there are 6*3*3*3*3 = 486 sets for part (e)

(f) First we choose which attributes are the same.  There are C(4,3) = 4 ways of choosing which attributes are the same.  There are then 3*3*4 = 36 ways to assign which is which for each of these three attributes, and there are 4 ways to assign the choices for the fourth attribute, so there are 4*36*4 = 576 sets for part (f).

sin (3x)   =  cos (6x)

3x + 6x    = 90

9x  =  90

x  = 10°  = theta

Note  that  we  can  write

log₁₆ 7^2  = a            (log property    log a^b =   b log a)           

2 log ₁₆ 7  =  a

log ₁₆  7   =    a/2        using the  change  of  base  theorem  (log  _a b =   log b / log a  )

log 7  / log 16  =  a/2

2log  7 / log 16   =  a

Also

log 2.5 / log 7  =   b 

So    ab     =    (2log 7)/log 16  *   log 2.5  / log 7 =      2log 2.5 / log 16  = 

2log 2.5  / log 4^2   =     2.og 2.5  /  [ 2log 4 ] =      log  2.5 / log 4

So.......

ab +  1  =      log 2.5  / log 4   +  log4 / log 4 =     [ log 2.5   +  log 4  ]  /  log 4  =

(log a +  log b  =   log (a*b)   )

log (2.5 * 4)  / log  4  = 

log 10  / log  4  =       

1/log  4    

log property      a^(1/log a)        =    10

4^(1/log 4)   =   10   

The sum is 4821.

Thank You so much 2sqrt37 was right Also how do you like something I can't figure it out

Thats HARSH

In rectangular form, 2*e^(pi*i/6) = sqrt(2) + i*sqrt(2).

Thanks so much this really helped :D

THX....working too fast....I'll correct  that  !!!!

1.   The  vertex  is    (-3, -5)

The axis of symmetry   is   x  =  -3

2.   To find the x intercepts

2x^2 + 13x  -  7   =    0 

(2x -1)  ( x + 7)   =   0

Set  both factors to 0   and solve  for  x  and we  get that    x =   1/2 =  q     and  x  =   -7   =   p

3.   Something left out  here

 The x-intercepts of the parabola y=x^2+bx are  ??? and  ???

The length of that type of diagonal is sqrt3 times the side length. So the rectangle would be \(x\sqrt{3}\) times \(x\) or \(x^2\sqrt{3}\)

The area of a hexagon is \((3\sqrt{3})*x^2/2\). What we have is \(x^2\sqrt{3}\). So multiply that by 3/2 to get 40*3/2=60