CPhill

Here's what I found....

The prime digits wil be  2,3,5,7

Cases :

Case 1 :  all the digits are the same  =  4  ways to do this 

Case 2 :  three of the digits are the same

      4 ways to select the same three digits  

      Then, select any one of the three remaining digits  to occupy any of  four places

      4  * 3 * 4  =  48

Case 3:  two of the digts are the same

     4 ways two select the same two digits 

     Choose any two of the remaining  three primes 3C2 = 3  to occupy any two of four positions 4C2 = 6

      4 * 3 * 6  =  72

Case 4 :  all the digits are different  =  4!  = 24

Total  =   4 + 48  + 72 + 24  =   148 

Do you mean                     ( 2x^4 -3)

                           y =          ____________         ????

                                          (x^2 - 4x + 1)

Area of quadrilateral = AD *DC =  180

Shaded area  (a trapezoid) =  (1/2) AD * ( DC + DC/3)  =  (1/2) AD *(4/3)DC = (2/3) AD * DC

So....the shaded area  =  (2/3) (180)  =  120 

https://web2.0calc.com/questions/problem-on-spheres

https://web2.0calc.com/questions/plz-help_20418

floor [ (2 -3x) / (3x + 8) ]

The value  of  f(1)  to f(1000)  always =   -1

So....the sum is  -1000 

Area =  6 (1/2) * 3^2 * sqrt (3) / 2  =

27 sqrt (3) / 2 ≈  23.38

Area of a hexagon =  6* (1/2) s^2* sqrt (3)/2      where s is  the side length

So

3 = 6 * (1/2) * s^2 * sqrt (3) / 2

3 = 3*  s^2 * sqrt (3) / 2

s^2 = 2/sqrt (3)

s = sqrt [ 2 / sqrt 3] ≈ 1.075

Note that the sum of the first n terms =   n * a1  +  d * (n)(n-1) / 2     where n > 1

So

S10  =  10a1 + d (10)(9) / 2  =   10a1 + 45d  =  0         (1)

S20 =   20a1 + d(20)(19) / 2  =   20a1 + 190d = 1/5     (2)

Multiply (1) by -2  and add to (2)  giving

100d = 1/5

d = 1/500

To find  a1

10a1 + 45 (1/500)  = 0

10a1 = -45/500

a1 = -45 / 5000  =  -9 / 1000

So

S70  =   70 (-9/1000) +  (1/500) (70)(69) / 2  =    21 / 5  =  4.2

log₂ (2x)  = log₄ (16 + x)

Using the change-of-base theorem

log (2x) / log2  = log (16 + x) / log 4

log (2x) / log 2 = log (16 + x) / log 2^2

log (2x) / log 2  = log (16 + x) / [ 2 log 2]      multiply through by log 2

log (2x)  = log (16 + x) / 2

2log (2x) = log (16 + x)

log (2x)^2 = log (16 + x)      this implies that

(2x)^2 = 16 + x

4x^2 - x - 16  =  0

Using the Quadratic Formula  (and taking the positive value of x )

x =  [ 1 + sqrt [ 1^2 - 4 (4) (-16) ] ] / (2 *4)

x = [ 1 + sqrt (257) ] / 8