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We can add 4 + 1 + 2 + 3 to get 10. So, for every 10 marbles, there are 2 marbles that are colored blue. for 20 marbles, there are 2 x 2 = 4 blue marbles.

Answer: 4 blue marbles

Well we have the same amount of members painting the wall, and if 40 members paint a 100 square foot wall in 50 minutes, then 40 members can paint a 480 square foot wall in 50 x 480/100 or 50 x 4.8 which is equal to 240 minutes or 4 hours.

Answer: 240 minutes

Thank you :D

Simplify  as

x^2  +  14x  +  31

Discriminant  =   14^2  - 4 (1)(31)  =  196 - 124   =   72

Simplify  as

-7x^2 - kx + 95   =  0

If this  has no real  solutions,  the discriminant  <  0

k^2 - 4(-7) (95)  <  0

k^2  + 2660 <  0

No integer  k  makes this  true

Circumference of  cone base =   ( 2 pi ) (36) (150 / 360)  =   30 pi

Radius of cone =    30 pi  / ( 2 pi )  =  15  =   r

Height of cone =  sqrt [ 36^2 - 15^2 ]  =  sqrt [ 1071 ]  =   3sqrt (119)

Volume of  cone =   (1/3) pi  r^2 * h   =   (1/3) pi * (15 )^2 *  3 sqrt (119)  = 

225 sqrt (119)  pi ≈   7710.9

Call the radius of the  sphere   = r  = the radius of the cylinder

Call the height of the  cylinder  =  2r  =  h

Volume of  cylinder  =   (4/3) pi r^3

144 pi  = (4/3) pi r^3

(3/4) 144 =  r^3

108   = r^3

(27 * 4)  = r^3

3(4)^(!/3)   =    r

Surface area of  cylinder  =  2 pi r^2  + 2pi r * h  =

2 pi [ 3(4)^(1/3)]^2  +  2 pi [( 3 )(4)^(1/3)] * [(2)  (3)(4)^(1/3) ]   ≈  427.48

Lateral surface area of the cone  =  pi * r * L      r = 7    L = 18

So   A =   pi * 7 * 18    =   126 pi   = 

Lateral surface area of the cylinder  =  2pi r *h  =  2 pi * 8 * 15 =  240 pi    = B

A /  B =       [126 pi ]   ) /  [ 240 pi]  =  .525

No rectangle  shown

We need to know the side lenght of the  triangle

-7 sqrt ( x - 1)  < -14                  divide both sides by -7 ,  reverse the inequality sign

sqrt (x - 1) >  2                          square both sides

x - 1  >  4

x >  5

Open circle at x = 5 with an arrow pointing to the right   ( to indicate that the the solution does  not stop at  x =10 )

This is AoPS homework.  Do not post AoPS homework.

Simplify as

-x^2  + 23x  + 7   =  0

x coordinate of vertex =   -b /(2a)  = -23 / (2 * -1) =   23/2 

y coordinate of  vertex =  -(23/2)^2 + (23)(23/2) + 7 =  (557/4)

Vertex  = ( 23/2  ,  557 / 4)

Simplify  as

x^2 - 2x  - 8   = 0

(x - 4) (x + 2)  = 0

x - 4  =  0                x +  2  = 0

x  = 4                      x =  -2

Simplify  as

7x^2 + 4x  + 36  =  0      divide through by 7

x^2  + (4/7)x + 36/7  = 0

x^2  + (4/7)x  =   -36/7       coplete the square on x

x^2 + (4/7)x  + 4/49  = -36/7 + 4/49

(x - 2/7)^2  =  -248/ 49          take both roots

x - 2/7 = [2sqrt (62) / 7] i

x =  2/7 + [2sqrt (62) / 7 ]  i            and   x =  2/7 - [ (2sqrt (62) / 7 ]   i

ab =  (2/7)(2/7)  =   4 / 49

sqrt  [ a +  sqrt b ] 

b  must be   1, 4 , 9 , 16, 25  or 36

b               a

1            3, 8 

4            2, 7

9            1, 6

16           5

25           4

36           3,10

(10 ordered pairs)

Sorry... that should be 

4 ( x- 2.25)^2   - 20 .25 + p           SO   p would need to be  + 20.25         Sorry for my error ! 

Thx, EP

Here's something else to  consider :  the x value of the  axis of symmetry = the x coordinate of the  vertex of  a quadratic  in x

x value  of  the vertex  =   -b /(2a)  =  0

The only way this can  = 0 is  if b = 0

Simplify to 

x^2 + 31x + k        now complete the square for 'x'

(x+ 15.5)^2    - 240.25   + k       will have a double root at x = -15.5   when  k = 240.25