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1. Since the balls are indistinguishable, we just need to keep track of how many balls in each box. 

The possible arrangements are:

\((4, 0, 0); (3, 1, 0); (2, 2, 0); (2, 1, 1) \Rightarrow \boxed{4}\)

2. We start by splitting the numbers into three cases:

The one digit numbers don't have any zeroes. 

The two digit numbers use two zeroes: 10 and 20. 

There are \(3^2 = 9\) three-digit numbers starting with 1, and 9 starting with 2. For each leading digit, a zero appears in each digit in \(9\div3 = 3\) of the numbers, so each has a total of \(3 + 3 = 6\) zeroes. Thus, the 3-digit numbers contain \(2\cdot6 = 12 \) zeroes. 

3. Here is the 3 by 3 grid:

For the first column, we have 3! = 6 arrangements of red, green, and blue. 

For the second column, we have only \(\binom21\) arrangements of red, green, and blue. 

For the last column, we have only 1 arrangements of red, green, and blue. 

For a total of 6 * 2 = 12 arrangements. 

I hope this helped,

Gavin.

(25x^3 - 10x^2 + x )

_______________         ≥ 0

      8 - x^3

x ( 25x^2 - 10x  + 1)

________________       ≥  0

(2 - x) ( x^2 + 2x + 4)

x ( 5x -1) ( 5x - 1)

_________________      ≥  0

  (2 - x) (x^2 + 2x + 4)

x (5x - 1)^2

________________        ≥  0

(2 - x)(x^2 + 2x + 4)

Note  that   (5x - 1)^2   is  ≥ 0  for all x   and  ( x^2 + 2x + 4)  is > 0  for all x

So   we only need consider what  x values makes

  x 

_____     ≥ 0

2 - x

Note that  if    x  is negative...the function is negative

And if  x > 2, the function is negative

So...the x values that make this ≥  0    are   0 ≤ x < 2 

I'm not that good at these, but I will give this one a try

Note that, for the given constraints, as small as the first function can be for any x  is  -4

So.....any value we choose for x  will be evaluated as being ≥ -4

So...in the first case....we are looking for  this

(x^2 - 4)^2  - 4  = 5

x^4 - 8x^2 + 16  - 4  = 5

x^4  - 8x^2 + 7  = 0

Factoring, we have

(x^2 - 1 ) ( x^2 - 7)  = 0

Setting both factors to 0 and solving for x  results in the values    1, -1 ,  √7 , - √7 

All of these make f (f(x))  = 5

In the other case, we need to choose an x < -4  which we can put into  x + 3  which will result in 3  or -3....note that when we put  -3 or 3 into the first function we get  5

0 will make x + 3  = 3  but x has to be < -4 to use this function

But  -  6  will work  because   f(-6)  =  (-6) + 3  =  -3

And this value when plugged into the first function to give a result of 5

That is  f ( f(-6) )  =  5

So...the values for x  that  will work are   1, -1 ,  √7 , - √7   and  - 6

At a 9% off sale Alexandra bought a new drill and saved $17. How much did she pay for the drill? (two decimal places)

Call the original price, P

9%  off  = 91%  of the original price  = .91P

And a savings of $17  = P - 17

So we have 

P ( .91)  = P -  17     add 17 to both sides,  subtract .91P from both sides

17  =  1P  - .91P

17  = .09P     divide both sides by .09

$188.88  = P  (the original price )

Richard's furniture store bought a couch and marked the price up 31%. At a 11% off sale the couch sold for $2,820. How much did Richard pay for the couch originally? (two decimal places)

Call the original price that Richard paid, P

The price after the markup was  (1.31) P

After an 11%  markdown the price was  ( 1.31 P ) (.89)

So we have

(1.31 P ) (.89)  = 2820

1.1659 P = 2820       divide both sides by  1.1659

P  = 2820 / 1.1659  =  $2418.73

The  population after a number of years  is given by :

Beginning population* (1 +  percentage growth rate expressed as a decimal)^(no. of years) =

48314  ( 1 + .035)^32  =   145266  people after 32 years

2x^2  + mx  + 8

If this has two distinctreal roots, then the discriminant  is  >  0

Sp

b^2  - 4ac  > 0

m^2  - 4(2)(8)  > 0

m^2 - 64  > 0

m^2  >  64

The possible values for m are

( - inf, -8)  U   (8, inf )

oH, i see, got it. Thanks i guess

Wait no i need exact

1. 89761*1063/1000 is about [95416] <---ANSWER

2. 107.96/100x=88652 x=[82115.59] <---ANSWER

No, that is NOT right!!

Remember you have 3 feet + 3 feet =6 Feet + width x 6 feet + Length

[156 x 56] - [150 x 50] =1,236 ft^2 - area of the deck!!.

1. 24/100*1501= 360.24 ANSWER: 360.24

2. 14/336=4.17/100. ANSWER: 4.17

3. 25700/69 is about 372.46 ANSWER: 372.46

The price goes up 8% a year so thats 0.08. The original price is $200.

200*(1+0.08)^20

The price will be about $932. (incredible iThing answer)

Find the whole area of the swimming pool plus the deck and then subtract the area of the swimming pool.

(50+3)(150+3)=8109.

8109-(50)(150)=609 ft^2 (Answer for pool deck area)

1)  $1,669 x [100% - 9%] =

     $1,669 x        91%         =

     $1,669 x         91/100   =$1,518.79 what Steven paid.

2)  $604 / [100% - 7%]   do as above

      $604 / [93/100]

      $604 / 0.93 =$649.46 - TV's price before 7% sale.

The answer is 52.

Let R1  be the radius of the first circle

And let R2 be the measure of the second circle

So

Arc length  =   radius * theta ( in rads )

So....since the arc lenghts are equal....

R1 * pi/2   = R2 * pi/3

R1 / 2  =  R2 / 3

R1  =  (2/3)R2

So...the area of  the first circle  is

[ pi * (R1) ^2 ]  =    pi *[  (2/3) R2 ] ^2  =  pi  (4/9) (R2)^2

And the area of the second circle  is

pi [ R2]^2  

So  the ratio  of the area  of the first cirlce to the second  is

[ pi * (4/9) (R2)^2 ]               4

_______________  =       ____

[ pi * (R2)^2 ]                        9

https://latex.artofproblemsolving.com/6/1/f/61f58bfedd7349150bfe2ec4ede4f440b8916f01.png

This is the "diagram", can someone please help?

Wolfram/Alpha's answer is "at least 3 fives, but it could be 4, 5, or 6 fives". The answer it gives is "minimum of 3 fives", which is what you want. Isn't it?

Doesnt it have to be 3, or 6 fives to be divisible by 125? Please GIVE ME A HINT

Note that the  "formula"  for determining the measure of an interior angle in any regular polygon  is

180 (n - 2) / n

Note that an integer value for an interior angle's measure will occur  whenever  n  divides 180  evenly

So regular polygons  with sides of 3,4,5,6 and 9  will have integer-valued interior angle measures

Also..... when n  = 8, the interior angle will measure  180 * 3/4  = 135°

So.... only the polygon with a side of 7 will not have an interior angle with an integer-valued measure

(4/3)x^2  + 4x + 1       factor out the 4/3

(4/3)  [ x^2  + 3x   +  3/4 ]       complete the square  on the quadratic inside the brackets

(4/3)  [ x^2  + 3x + 9/4  + 3/4  - 9/4 ]     factor the first three terms and simplify the rest

(4/3) [ (x + 3/2)^2  - 3/2 ]     distribute the (4/3)  over the terms

(4/3) ( x + 3/2)^2    - 2    =   (4/3) ( x + 3/2)^2  + (-2) 

So...in the form   a( x +b)^2 + c

a = (4/3)   b = (3/2)   and  c  = -2

So

abc  =  ( 4/3)(3/2)(-2)   =  -24 / 6   =   -4

log_a s  = 5    log_a t   =10

In exponential form, we have

a^5  = s       and  a^10  =  t

So

log_a (s / t^2 )  =

log_a [a^5 / (a^10)^2 ]  =

log_a [ a^5 / a^20 ]  =

log_a  a^(-15)        and by a log property we can write

-15 * log_a a  =

 ( Note  log_a a   =  1  )

-15 * 1  =

-15

It can be either. 

Here is an example.

\(x^2+6x+5 = (x+5)(x+1)=(x-(-5))(x-(-1))\)

If p and q are positive numbers then it will be +

If p and q are negative numbers then it will be -

That is assuming that b and c are both positive numbers.

Anyway it is all related to the signs of all the pronumerals.    b,c,p and q

Ah, no. I dont look at the answer unless it really is required, given that i dun even know where to get the answer