geno3141

For real numbers:

x⁴ - a⁴  =  (x² + a²)(x² - a²)  =  (x² + a²)(x + a)(x - a​)

For complex numbers:

x⁴ - a⁴  =  (x² + a²)(x² - a²)  =  (x+ ai)(x - ai)(x + a)(x - a​)

Since  f(x)  =  2, there are two ways that the answer is 2.

First way:       if x < 2:   9x + 16  =  2   --->   9x  =  -14    --->   x  =  -14/9.

Second way:  if x >= 2:  2x - 14  =  2   --->   2x  =  16     --->   x  =  8

You will now need to add the two possible values for x ...

You've started correctly ...

You'll need to factor x² + 11x+ 30.

x² + 11x+ 30  =  (x +    )(x +   )

What two numbers add to 11 and equal 30 when multiplied?

1)  Since OA is perpendicular to DAE, you can calculate the size of angle(BAO)

[In the following statements, the reason that the triangles are isosceles is because

two of their sides are radii.]

2)  Since triangle(BAO) is an isosceles triangle and you know the size of angle(BAO),

     you can calulate the size of angle(OBA) and then the size of angle(BOA).

3)  Since triangle(BOC) is an isosceles triangle and you know the size of angle(CBO),

     you can calculate the size of angle(BCO) and then the size of angle(COB).

4)  Since you know the sizes of angle(BOA) and angle(COB),

      you can calculate the size of angle(COA).

5)  Since triangle( COA) is an isosceles triangle and you know the size of angle(COA),

      you can calculate the size of angle(OAC).

If you have two solids that are similar,

-- the ratio of the lengths equals the ratio of the square roots of the areas

-- the ratio of the areas equals the ratio of the squares of the lengths

-- the ratio of the lengths equals the ratio of the cube roots of the volumes

-- the ratio of the volumes equals the ratio of the cubes of the lengths

Therefore for similar figures A, B, and C:

                       A       B        C 

lengths   =      3       4         5

areas      =      9     16       25

volumes  =    27     64     125

x + y  =  10     --->     x  =  10 - y

x³ + y³  =  460     --->     (10 - y)³ + y³  =  460

            (1000 - 300y + 30y² - y³) + y³  =  460

                             30y² - 300y + 1000  =  460

                               30y² - 300y + 540  =  0

                                       y² - 10y + 18  =  0

Using the quadratic formula:  y  =  5 + sqrt(7)     y  =  5 - sqrt(7)

x + y  =  10     --->     x  =  10 - y   

If  y  =  5 + sqrt(7)      --->     x  =  10 - ( 5 + sqrt(7) )     --->     x  =  5 - sqrt(7)

If  y  =  5 - sqrt(7)       --->     x  =  10 - ( 5 - sqrt(7) )      --->     x  =  5 + sqrt(7)

( 5 - sqrt(7), 5 + sqrt(7) )    and     ( 5 + sqrt(7), 5 - sqrt(7) ) 

f(x)  =  x5 + 8x4 +2x2 + 5x + 15

1)  You'll need to find the anti-derivative of f(x) 

--->     F(x)  x6/6 +8x5/5 + 2x3/3 +5x2/2 +15x + c

2)  You'll need to evaluate it from -6 to 6

--->   F(6) - F(-6)

--->   [ (6)6/6 +8(6)5/5 + 2(6)3/3 +5(6)2/2 +15(6) + c ] - [ (-6)6/6 +8(-6)5/5 + 2(-6)3/3 +5(-6)2/2 +15(-6) + c ]

3)  Finish ...

I'm assuming that the function is  f(x)  =  -2x² + 32x + 5.

To divide the interval (0,16] into four rectangles, the dividing points are:  0,  4,  8,  12,  16

The left-hand endpoints are: 0, 4, 8, and 12.

f(0)  =  5

f(4)  =  101

f(8)  =  133

f(12)  =  101

To find the areas of the rectangles, multiply each height by its width (which is 4).

5x4  +  101x4  +  133x4  +  101x4  =  1360

x² + 6x + y²  =  4

Complete the square for x:  ( x² + 6x + 9 ) + y²  =  4 + 9

Factor:                                           (x + 3)² + y²  =  13

The problem has now become:  what two perfect squares have a sum of 13?

--  they are 4 and 9.

So, either  (x + 3)²  =  4   and   y²  = 9

--->                x + 3  =  2             y  =  3

                            x  =  -1            y  =  -3

              or    x + 3  =  -2

                           x  =  -5   

              Combining these answers to get:  (-1,3),   (-1,-3),   (-5,3),   (-5,-3)

Or:   (x + 3)²  =  9   and   y²  = 4

--->                x + 3  =  3             y  =  2

                            x  =  0            y  =  -2

              or    x + 3  =  -3

                           x  =   -6   

              Combining these answers to get:  (0,2),   (0,-2),   (-6,2),   (-6,-2)

If it is equally likely that each passenger can stay at any of the three hotels:

--  the first passenger can stay at any hotel   --->   4/4  =  1

--  the probability that the second passenger stays at the same hotel   --->   1/4

--  the probability that the third passenger stays at the same hotel   --->   1/4

If these are independent choices:  the probability is 1 · ¼ · ¼  =  1/16.

[However, all three persons could be from the same family, which means that the choices are not independent.]