geno3141

x³  =  -4 + 4i

Step 1)  write  -4 + 4i  in  r·cs( theta ) form

         r  =  sqrt( (-4)² + (4²) )

         theta  =  tan^-1( 4 / -4)           

                                                        that your angle is in the second quadrant>

Step 2)  To get your first answer:

                  take the 3rd root of the r-value and divide the angle by 3

Step 3)  To find the other roots:  

              --  add 2pi/3 to the angle of the first answer (keep r the same)

              --  add 2pi/3 to the angle of the second answer (keep r the same)

You can use this formula to help you find the number of terms:  t_n  =  t₁ + (n - 1)d

t_n  is the last term of an arithmetic sequence.

t₁  is the first term of an arithemtic sequence

n   is the number of terms

d   is the common difference

This sequence contains two arithmetic sequences:

6, 10, 14, 18, ... 94, 98          and        7, 11, 15, ... 91, 95

In both of these sequences, d = 4.

For  6, 10, 14, 18, ... 94, 98                                   t_n  =  t₁ + (n - 1)d

t_n  =  98                                                                 98  =  6 + (n - 1)4

t₁  = 6                                                                    92  =  4n - 4

n   =  unknown                                                       96  =  4n

d    =  4                                                                   24  =  n

For  7, 11, 15, ... 91, 95                                         t_n  =  t₁ + (n - 1)d

t_n  =  95                                                                 95  =  7+ (n - 1)4

t₁  = 7                                                                    88  =  4n - 4

n   =  unknown                                                       92  =  4n

d    =  4                                                                   23  =  n

So, there will be a total of  24 + 23  =  47 terms

For the function:  f(x)  =  2x -5:

First, find the inverse:

step 1)  replace "f(x)" with "y":        y  =  2x -5

step 2)  interchange x and y:          x  =  2y - 5

step 3)  solve this equation for y :   x  =  2y - 5

                                                  x + 5  =  2y

                                             (x + 5)/2  =  y

                                                         y  =  (x + 5) / 2

step 4)  replace "y" with "f^-1(x)":  f^-1(x)  =  (x + 5)/2

This equation is the inverse.

Now, solve the problem:  For what value of x will f(x) = f^-1(x)?

Set  f(x)  =  f^-1(x)     --->     2x - 5  =  (x + 5)/2

                  4x - 10  =  x + 5

                       3x - 10  =  5

                                                  3x  =  15

                                                    x  =  5

If  g(x)  =  2x² + 2x -1,  find  g( g(2) ).

Work from the inside out; find the value of g(2).

    g(2)  =  2(2)² + 2(2) - 1  =  11

Now, replace the g(2)  of g( g(2) )  with 11,

so you now have  g( 11 )  =  2(11)² + 2(11) - 1  =  263.

To summarize:  g( g(2) )  =  g( 11 )  =  263

The graph of  y  =  x² + kx + 1  is a parabola with its minimum point at its vertex and rises, on both sides,  from there.

There is no way that this parabola will always be below zero; that is, will always be below the x-axis. So, the answer is the null set.

If the graph were:  y  =  - x² + kx + 1, which is a parabola with a maximum point and falling, it would be possible for the graph to be always below the x-axis.

Draw square MNPQ.

Find the midpoint of MN, call it "X".

Using X as the center, draw the semicirce inside the circle that starts at M and ends at N.

Choose a point Y on the semicircle. Angle(MYN) will be a right angle because it is inscribed

in a semicircle.

If you choose a point Y' inside the semicircle, angle(MY'N) will be an obtuse angle.

If the point that you choose is outside the semicircle, it will be an acute angle.

Now, to find the probability of its being acute.

We need to compare the area outside the semicircle to the area of the whole square.

Assume that the square has a side length of 2.

Then, the area of the square is 4.

The area inside the semicircle will be:  ½·pi·r2  =  ½·pi·(1)2​  =  ½·pi.

The area outside the semicircle will be:  4 - ½·pi.

The probability will be the area outside the semicircle divided by the area of the square:

   probability  =  ( 4 - ½·pi ) / 4   

[Because we're finding a ratio, we can assume whatever we wish for the side length of

the square. If you are in doubt, choose another number for this length, and recalculate.]

The first term is 1 and each successive term is the sum of all the previous terms:

1, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192

Although not true for the first term, each of the following terms has a value of 2^{n - 2}

where n is the number of the term.

We want       2^{n - 2}  >=  5000

              log(2^{n - 2})  >=  log(5000)

          (n - 2)log(2)  >=  log(5000)

                      n - 2  >=  log(5000)/log(2)

                      n - 2  >= 12.29

                            n  >=  14.29

So:  n  =  15

I believe that the number is 198.

Using these digits, two at a time (but there is no restriction that the same digit isn't used twice), the possibilities are:

11,18, 19, 81, 88, 89, 91, 98, 99

The sum is 594.

Dividing this by 3, gives us i98.

If  x = 3  then  | (3)² - 16 |  =  | 9 - 16 |  =  | -7 |  =  7

If  x = -3  then  | (-3)² - 16 |  =  | 9 - 16 |  =  | -7 |  =  7

-3 x 3  =  ...

If he never goes more than two days in a row without running, the minimum number of days that he runs is 4.

The schedule looks like this: NNYNNYN NYNNYNN

If he never runs more than two days in a row, the maximum number of days that he run is 10.

The schedule looks like this:  YYNYYNY YNYYNYY

[Notice that the schedules are the exact opposite.]

10 - 4  =  ????