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[quote:Melody]  $ u^2=1+sin\theta $  [quote]

how about just $u=\sin(\theta)$ ?

its raised in the power of two from the thing itself

and i do not see how you would get that $1$ except when applying the common derivative

care explaining? :)

that gives us $ x_1=\frac{10+2\iota }{2\cdot \:1}  \ \ \ \Rightarrow \ \ \ \frac{10+2\iota }{2}   \ \ \ \Rightarrow \ \ \ \frac{\cancel{2}\left(5+\iota \right)}{ \cancel{2}}  $  

finally, we get 

$  x_1=5+\iota$

fixed.

i would recommend doing b, then going on with a;

though, before we continue any further, lets just work the function a bit:

$M(s)=-\frac{1}{28}(s^2)+(3+8)s+(31+8)$

$M(s)=-\frac{1}{28}s^2+\left(3+8\right)s+31+8$

$  M(s)=-\frac{1}{28}s^2+11s+39  $

used desmos and got this image:

now its asking us for the cars best gas mileage, and the speed its in

in our graph, the x axis is going to be speed, and the y axis is going to be the gas mileage

thus, by looking at the coordinate of the vertex, or the max, we can see that the best mileage is at $\boxed{886 \ \text{miles per gallon}}$, whilst the speed this is attained is at $\boxed{154 \ \text{miles per hour}}$.

:D
 

her eis the image as well so you have an idea of what we are doing

the formula for finding the area of a circle sector is:

$ \mathcal{A}= \frac{\theta }{360}  \times \pi r^2 $ where $\mathcal{A}=$Area    ;   $\theta=$given angle  ;  $r=$radius  

we are given the value of area and theta. so lets substitute that in:

$ 250 \pi= \frac{100}{360}  \times \pi r^2  $

you can cancel the $\pi$s, so you get 

$ 250= \frac{100}{360} \times  r^2 \ \ \overset{ simplify }{===== \Rightarrow} \ \ \ 250 = \frac{5}{18} \times  r^2    $

$  5r^2=4500  $

$ r^2=900  $

$ r=\pm \sqrt{900}  $

        $\Updownarrow  $

$ r=30$  and  $r=$-30 

only the positive one is valid, thus, the radius of that circle is $30$. 
 

$x^2+29=10x+3$

take the 3 to the left-hand side

$  x^2+26=10x $

now take the 10x to the other side

$  x^2-10x+26=0 $

now to solve this you can pick many ways, though i will pick using the quadratic formula

$ _2x_1=\frac{-b\pm \sqrt{b^2-4ac}}{2a}  $

where $a=1$  ;  $b=-10$ ;  $c=26$

just plug and chug

$ _2x_1=\frac{-\left(-10\right)\pm \sqrt{\left(-10\right)^2-4\cdot \:1\cdot \:26}}{2\cdot \:1} $

$ _2x_1=\frac{-\left(-10\right)\pm \iota \sqrt{10^2-104}} {2\cdot \:1} $

$ _2x_1=\frac{-\left(-10\right)\pm \iota \sqrt{4} } {2\cdot \:1} $

$ _2x_1=\frac{-\left(-10\right)\pm \:2\iota }{2\cdot \:1} $

that gives us $ x_1=\frac{10+2\iota }{2\cdot \:1}  \ \ \ \Rightarrow \ \ \ \frac{10+2\iota }{2}   \ \ \ \Rightarrow \ \ \ \frac{\cancel{2}\left(5+\iota \right)}

{\cancel{2}}  $  

finally, we get $x_1==5+\iota $

second root will be as following:

$x_2=\frac{10-2\iota }{2\cdot \:1}  \ \ \ \Rightarrow \ \ \ \frac{10-2\iota a}{2} \ \ \ \Rightarrow \ \ \ \frac{\cancel{2}\left(5-\iota \right)}{\cancel{2}}$

thus the second root is $  x_2=5-\iota $

The statement you're trying to prove is obviously false and it's false where you said you checked.  Take n = 0.  Then $5^{n^3}+1 = 5^0+1 = 2$.  Also, $3^{n+1} = 3^1 = 3$.  Obviously 2 is not divisible by 3.
 

This question has been asked before.  See a solution at https://web2.0calc.com/questions/polynomial_80#r1.  You may also want to see https://web2.0calc.com/questions/hard-algebra_17#r1 which solves this kind of problem but it has more explicit details.   If you don't understand some steps, ask.

Ignore the nulls so that P(R) = P(B) = 1/2.

Calculate the chances of you losing 59 times in a row and Compare these chances to the odds of winning the NJ lotto (say 1/150M). risk. Yes, you calculated the numbers correctly and you have an edge. But there are other issues to consider before deciding whether to play. The most significant one is “gambler’s ruin”: eveb though you have the edge and would surely win if you can play forever (Strong Law of Large Numbers), you can easily go broke if you have insufficient capital.

Chance of losing 59 times in a row is   1/(2^59) = approx   1/ 5.76*10^17

1/(150M)   I will assume M means million.    is approx   1/1.5^10^8

You are much much much more likely to win with lotto.  Stick with lotto.  This place is rigged!

$\int \frac{sin^2\theta \:cos\theta }{\sqrt{1+sin\theta }}\;d\theta$

let   $u^2=1+sin\theta$

If you need more help then show us what you have managed to do on your own. 

------------------------------------------------

Ps.   I have done this almost to the end. Unless I have made some really silly mistake/s, it appears to work fine.

Options 

Red shirt, pants, green hat = 5*6*8 = 230 possibilities

or

Green shirt, pants, red hat = 5*6*8= 230 possibilities

Total = 230+230 = 460?

i got that :P

So if this is not answered 'soon'  you will no longer be interested in the answer?

So you do not want to understand you just want the answer?  ok, Got it.

Understanding takes time. 

When you research and work it out for yourself the understanding will usually be deeper.

Of course, if you just want the answer, that is called cheating.

Why is it urgent?

Sounds to me like your teacher really wants you to think hard and do some research on your own.

(by Vieta's rules)

r+s = -6   <======   r^2 +2rs + s^2 = 36       r^2 + s^2 = 36-2rs

rs = -2   <====== 

(r-s)^2 = r^2 -2rs + s^2    

             r^2 + s^2  - 2rs

              36-2rs - 2 rs

               36 -  -4  -  -4    =  44      

Can you say it in words?

This question has been asked before.   See a solution at https://web2.0calc.com/questions/hard-algebra_17#r1

The discriminant of the Quadratic Formula must = 0 for this to occur

b^2 - 4ac = 0

b^2 = 4ac

b^2 = 4(2)(82)

     b  =  +- sqrt (656)

Try using the quadratic equation. 

=^._.^=