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First, we try to factor x^2 + 2x + 10. That doesn't seem quite possible, so now we try to use completing the square with (x-1)^2. We get that x^2 + 2x +10 = (x-1)^2 + 4x + 9. We can still factor it more and get (x-1)*(x-1 + 4) + 13 = (x-1)*(x+3) +13. Hence, the quotient is x+3 and the remainder is 13.

ohhhh okay thanks for clarifying 

for some reason I was thinking that it meant and? 

or is it only for special circumstances? or am i just going nuts LOL

and so.... 
when describing more thanTWO continuities, intervals of increase, decrease? do we use union as in or? 
is that what they mean 

let's say we have an int. of decrease thats (-5,6)u(8,9) something like that did i do that right?? 

and I'm SO sorry i keep asking questions it's probably getting annoying. im sorry. 

U   =   "or" = "union"

∩  =  "and" = "intersection"

I have a quick question if that's okay though,,,it has nothing to do with this.... 


does the U for union represent and or or? 
if yes it is and, then what is the symbol for or? if no for and and it is or, then what is the symbol for and? 

The maximum frictional force will be   c_sf  m g

  If the force is greater or equal to this, the mass will slide

c_sf m g  =  ma     This 'a' is the decelleration max magnitude when the mass , m , will slide

c_sf g = a

.75 (9.81) = 7.358 m/s^2  DEceleration

Nope!! 
You did great, thank you so much :D

See full solution No. 6 here:  https://web2.0calc.com/questions/help_68853

I believe that Mr./Ms./Mrs. Soapen here has deleted his question. Not exactly a nice thing to do for the other people who want to learn from the answer and the question, but I suppose it's just his/her preference for people not to learn...

We can make a table:

1   2   3   4   5   6   7   8

                                                                                         1  1   2   3   4   5   6   7   8

                                                                                         2  2   4   6   8  10 12 14 16

                                                                                         3  3   6   9  12 15 18 21 24

                                                                                         4  4   8  12 16 20 24 28 32

                                                                                         5  5  10 15 20 25 30 35 40

                                                                                         6  6  12 18 24 30 36 42 48

We can see that in the table there are 48 values, and 24 of these are a multiple of three, so the probability that the product of the two rolls is a multiple of 3 is 24/48 = 1/2.

Thanks Melody. I most certainly did by using this very nice calculator that solves "trapezoids":

https://rechneronline.de/pi/isosceles-trapezoid.php

Thanks guest.

Here is the pic.

Did you get some program to work that out?

OK....I hope I didn't confuse you too much.....LOL!!!!

OHHHHH, 
oh my god CPhill  

THANK YOU SOOOO MUCH 

you don't know how much you and hectictar just helped me...i appreciate it so much thank you thank you thank youuuuuuuu 

1)

x^2 + 6x + 9  = 0

x^2 + 6x  =  -9

Take (1/2) of 6  = 3....square it .....= 9......add to both sides

x^2 + 6x + 9  = -9 + 9

x^2 + 6x + 9  =  0  .....notice that this is what we started with!!!.....factor the left side

(x +3)^2  =  0        take the square root

x + 3  = 0              subtract 3 from both sides

x  = -3

2) 

2x^2 + 6  = 24       subtract 6 from both sides

2x^2  = 18             divide both sides by 2

x^2  =  9                take both roots

x  = ±√9

x = ± 3

Note that  your graph  is  continuous  from negative infinity to  - 4.....but we have an open circle at  x  = -4

So.....if your graph stopped there  the interval of continuity would be (-infinity, -4)

However.....the "closed"  circle  at x  = -4  on the next function "fills in"  the continuity  which extends to "almost"  x  = 4

At x  = 4  the continuity stops

So.....your graph is actually continuous  from  (-infinity, 4)

i'm confused though,,,, 

how does

\((-\infty,4)\) include \([-4,4)\)? is there a method of knowing this? :( Im sorry im really confused ahhh

[ -4, 4)   is included in the interval  (-infinity, 4)

thanks CPhill! 

but how come during the inverval of continuity -4 isn't included? 
like wouldn't [-4,4) be an interval of continuity? How come that's not included? 

x + y + 3z = 10    ⇒ -2x - 2y - 6z  = -20      (1)
-4x + 2y + 5z = 7        (2)

kx + z = 3       (3)

Add (1)  and (2)   and we get that

-6x - z  =  -13      Multiply through by  -1

6x + z  = 13       (4)

Note that, in (3) , if  k = 6 we have that

6x + z  =  3      (5)

But this is impossible because  if 6x + z  = 13 in (4), it cannot also = 3   in (5)

So....k = 6   gives no solutions

Equation of line  with a constant increase  

We have points (-4,3)  and  (4,8)

Slope  =  [8-3] / [ 4 - -4]  =  5/8

Equation is

y = (5/8)(x - 4) + 8

y = (5/8)x - 20/8 + 64/8

y = (5/8)x + 44/8

y = (5/8)x + 11/2

Equation of line with constant decrease

We have points  (4,4)  and  ( 9,1)

Slope  =  [ 1 - 4] / [ 9 - 4]  = -3/5

Equation is

y = (-3/5)(x - 4) + 4

y = (-3/5) + 12/5  + 4

y = (-3/5)x + 32/5

Based on your graph, we would have this  :

                  1                          if    -inf  < x  -4

f(x)  =       (5/8)x + 11/2         if    -4 ≤ x < 4

                 (-3/5)x + 32/5       if     4 < x  ≤ 9

1) and 2)

This is still "piecewise"......if we can draw a graph without lifting our pencil, it is continous (although it could still be  piecewise-continous).....your graph is "broken"...so...it is not piecewise continuous...

3)

This graph is continuous from    (- infinity, 4)

     It is also continuous from  (4, 9 ]

Note that at x  = 4....the continuity is "broken"

Also note that the closed circle at x  = - 4   "picks up" the continuity from negative infinity

Hectictar can probably develop this even more   !!!!

f(x)  = x

ax  = (x + 1) x

ax  = x^2 + x

x^2 + ( 1 - a)x  = 0

x ( x + (1 - a) )  =  0       ⇒   x  = 0    (reject)   or

x + (1 - a)  =  0

x  = - (1 - a)

x   =  a - 1

x + 1  =  a

This must be correct because

 ax              (x + 1)x

____  =     _______   =      x

x + 1            x + 1

Aw thank you Melody, Tommarvoloriddle, and CPhill!! 
You guys are awesome!

and I'm not the one that really did anything,,, it was mostly Hectictar and her awesome explanation and clarifications which truly did help me and I appreciate her for it. :) 

:D

 

Oh my god, 

you are an absolute life saver hectictar, and i hope you know it :D 

Thanks to you, I can now do these without hesitations!! 

but if it's okay may i ask a few more questions? 

Alright,,,

So I have constructed the graph below and I don't understand the questions that i have written "on" the graph. If I could just get these questions answered, I will finally fully know this material :') 



Again, I just want to thank you so so so sooo so much for your help. I really appreciate it :D 

Let the intersection  of BO and AC  be  E

In triangle ABE....angle BEA  = 90°  angle BAE  =  angle BAC  = 28°

So....angle ABE  =  90 - 28  =  62°

And  the sum of  angle BAC  +  angle  OAC  =  28 + 32  =  60°

Therefore....in triangle BAO....angle AOB  = 180  - 62 - 60  = 58°

And by SAS,,,,triangle AEO  is congruent to triangle CEO

So...angle AOB  =  angle AOE  = angle COE  =  58°

So...angle AOC   =  angle AOE  + angle COE   =  58  + 58   =  116°