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\( \[ x^2 + \dfrac{5}{3}x + \boxed{\phantom{000}}. \]\)

Take   1/2  of 5/3  =  5/6  ....square it   =  25/36

So

x^2  +  5/3x  +  25/36      factors as

(x  + 5/6 )^2

So.....  25/36 is what we need

1.  The period  is  10  (  units )

2.  The minimum is -4 

3.  The maximum is 5

4.  The midline is   y  =  0

5. The amplitude is 5

There are 36 possible outcomes

The number of rolls where Amaya's roll is greater than Bo's  is

5  +  4  +  3  +  2   + 1  =  15

So....the probability is  

15 / 36   =

5  / 12

\(\triangle MNP\sim\triangle MAB\) because a segment located in the interior of the triangle is parallel to a side. You could also prove similarity by Angle-Angle Similarity Theorem. 

Using the above similarity statement, one can create a proportion because each side is proportional. The one I will use is \(\frac{MA}{MB}=\frac{MN}{MP}\). This comes from the similarity statement. 

Although we do not know the length of \(\overline{MA}\) directly, we can find it by subtracting the length of \(\overline{AN}\) from the length of \(\overline{MN}\). When we plug in these numbers, we can then solve for the unknown side length.

[ x / 6]  +  2  =  15          Given

[ x / 6 ]  + 2 - 2 =  15 - 2                 Subtraction Property of Equality

[x/ 6]  =  13

6 [ x/ 6]  =  6 * 13                        Multiplication Property of Equality

x  =  78

You fixed it. Thx i know. I dont want to  be complaining rather helping. its a struggle i can tell. thx for all the help! :-)

I'm just going to try enclosing the quadratic formula in dollar signs and see if that helps.

If if helps for me, which I think in might, does it cause display problems for other people?

Let me know.

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a} \qquad \text{No dollar signs}\\ \)

\($x = {-b \pm \sqrt{b^2-4ac} \over 2a} \qquad \text{Added dollar signs}$ \)

I have also discoverend that I do not need to go into LaTex box to do this, I can just type it straight in!

$x=\frac{1}{2}$   this is just typed straight in, now I am going to see what happens.

It worked perfectly!!

Fixed now....I'm trying to answer too many questions at once....LOL!!!

as a side note both problems i needed help with you got wrong but only because of little mistakes. Maybe pay a little more attetion? thx for your help i just needed some and i figured it out anyway. I hope that feedback helped

i figured it out you simplified it wrong

These are common reference angles,  Adam

You really just need to memorize this chart :

http://www.analyzemath.com/trigonometry/special_angles.html

Note that the reciprocal  of  √2/2  = 2 / √2  =  √2*√2 / √ 2 =  √2

It's not as bad as it looks...since  csc, sec and cot  are just reciprocals of sin, cos and tan, respectively.....just learn the values for sin, cos and tan......"flip" these over for  csc, sec and tan!!!

This is the way I  had to learn them.....  

that was incorrect but i dont know why

Can you show me the process of how you did at least one of them? Like, how did you jump from sin(pi/3) to √3/2?

Boy-Boy  =  C(5,2)  =  10 games

Girl-Girl = C(5,2)  =  10 games

Boy-Girl  =  5 * 5  =  25 games

So..... Boy - Boy  =

[ 10 ]  /  [ 10 + 10 + 25 ]  =

10 / 45  =

2 / 9

sin (pi/3)  =  √3/2

cos(pi/4) =  √2/2

tan (pi/4)   =  1

csc(pi/3)  =  2/√3  =  2√3 / 3

sec (pi/3)  =  1 / cos(pi/3) =  1 / (1/2)  =   2

cot (pi/6)  =  1 / tan (pi/6)  =  1 / (1/√3)  =   √3

P  +  .04* P   = 

P( 1 + .04)  =

P (1.04) =

(1.04)P

Note that  we have 5 girls....and we want to choose any 2 for a game....so

C (5.2)  =  10  games

-7 pi    is the same angle as  + pi ......  on the unit circle, the coordinates  are  (-1, 0)

cos(−θ) =−3/5       tan θ  > 0  

Note  that  cos(-θ) = cos (θ)  ....so....this angle is is in the 3rd quadrant.......so....the sin (-θ) is

y / r  =   - √ [5^2  - (-3)^2 ]  / 5   =  - √ 16/ 5   =   - 4 / 5

1. A population of bacteria doubles every 12 hours. Initially, the population of bacteria is 80.

What is the population of the bacteria after 30 hours?

We have the following

80 (2) ^t/12     ....so.......when t  = 30....we have

80 (2) ^30/12 ≈   453

2. What type of exponential function is f(x)=0.75(2.1)^x

In the form

f(x)  =  a* b^x      if  b >  1   we have a growth function  as x increases

So......This is a growth function......the rate of change  is  2.10 * 100  =   210 %

3.  Just  put 4 into the function

(4)^4−2(4)^3+5(4)^2−20(4)−4   =

256  - 128 + 80   -  80  - 4   =  

124

You're welcome  !!!!

(x^4y^7)^2/3  =

 ( [ x^4 y^7] ^2 ) ^1/3   =

( x^8 y^14 ) ^1/3  =

∛  [  x^8  y^14 ]  =

To determine  the powers that come out and those that stay in, do this

8/3  =   2  + 2/3          2 powers of x come out, 2 stay in

14/3  =  4  + 2/3         4 powers of x come out, 2 stay  in

x^2 y ^4  ∛  [x^2  y ^2 ]   

       .

Sweet, thanks for showing all the steps. Really helps me understand.

x^2  +  bx +  16

If it has one (repeated) root  then  the discriminant  is  = 0

b^2   -  4(1)(16)   =  0

b^2   -  64  =  0            add 64 to both sides

b^2  =   64                    take both roots

b  =  ±√64

b  = ±8

If it has two  roots, then the discriminant  is >  0

b^2  -  64  >  0

b^2  > 64

So.....   b < -8    or  b >  8

In interval notation, we have     (-inf, -8)   U  (8, inf)

The correct  equation in the first part should be

A  =   100(1/2)^t/5.27   

To  see that this is correct....let  t  =  5.27  and we have that

A  = 100(1/2) ^5.27/5.27   =  100  (1/2)¹  =  100 (1/2)   =  50%  left  after 5.27 years, the half-life  

Your   second answer is correct

I can't tell exactly what the third one is....sorry....!!!