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Btw, do you like my new pfp?

OK.....thanks for your explanation, Melody  !!!

That's good. 

OK....!!!

Thanks, CPhill! How's life been for ya?

Hi Chris,

Your answer is better than mine in that I certainly did not need to get into permutations.

My logic did lead to a correct answer I think (as verified below) but it made it unnecessarily messy and there was far too much room to make a numeric error.   

However:

Your answer is not correct because you have not included all the possibilities.

I'll try again. Using your ideas.

Ways to get all 5 from the same suit = 4*13C5    (Just like Chris said)

Ways to get 4 from one suit and 1 from another = 4*13C4*  39  (you forgot this one Chris)

Ways to get 3 from one suit and 2 from another = 4*13C3*   3*13C2

Total number of ways to choose 5 cards from 52 = 52C5

nCr(52,5) = 2598960 

Ways to get no more than 2 suits = 4*13C5   +  4*13C4*  39  +  4*13C3*   3*13C2

4*nCr(13,5)+4*nCr(13,4)*3*13+4*nCr(13,3)*3*nCr(13,2) = 384384

Number of ways of NOT getting 2 suits = 52C5 - 384384

nCr(52,5)-384384 = 2214576

So 

P(drawing at least 3 different suits) = 2214576 / 2598960

2214576/2598960​ = 0.8521008403361345 = 507/595

Which is exactly the same as I got doing it with permutations !!   

WOW 

Good to see you, RedGlasses !!!

Very nice, hectictar...!!!

Thank you so very much, hectictar. I was really stuck on that problem. You're a life saver. :D

Let's call the two numbers  a  and  b .

The sum of two numbers is 361.

a + b  =  361

The difference between the two numbers is  173.

a - b  =  173

Now we have two equations and two variables:

a + b  =  361

a  - b  =  173     We can solve the system of equations like this...

361 + 173   =   361 + 173            Plug in  a + b  for  361  since they are equivalent,

                                                    and plug in  a - b  for  173  since they are equivalent.

(a + b) + (a - b)   =   361 + 173

a + b + a - b   =   534

2a   =   534

a   =   267

Now we can use this value of  a  to find  b  .

267 + b   =  361

b   =   94

So the two numbers are  267  and  94  .  

1.  The first graph is just the sine function shifted down two units

Here's  the graph of both :   https://www.desmos.com/calculator/pl7ix8zvkb

2.. This is just the cosine graph  with a period 1/2 as long as  the parent function and shifted up by 1 unit

Here's the graph of both: https://www.desmos.com/calculator/nokkptgkli

3.  A little difficult, but here's one possibility...

In  the first second  the pendulum  swings to the max position (to the right) 

In the next second, it returns to its neutral position

In the third second, it swings to min position (to the left )

In the last second, it returns to the neutral position again

So....the function is

y  =  6sin (pi * t / 2)

Here's the graph : https://www.desmos.com/calculator/0cslmqnaad

\(f(\,g(x)\,)\,=\,\frac{3(\,g(x)\,)^2-\,10(\,g(x)\,)\,-\,25}{g(x)\,+\,1}\)

f( g(x) ) is undefined when  g(x)  is undefined, and  g(x)  is undefined when...

3x² + 11x + 10  =  0

3x² + 6x + 5x + 10  =  0

3x(x +  2) + 5(x + 2)  =  0

(x + 2)(3x + 5)  =  0

x  =  -2    and   x  =  -5/3

f( g(x) )  is also undefined when 

g(x) + 1  =  0

\(\frac{3x^2-10x-25}{3x^2+11x+10}+1\,=\,0 \\~\\ \frac{3x^2-10x-25}{3x^2+11x+10}+\frac{3x^2+11x+10}{3x^2+11x+10}\,=\,0 \\~\\ \frac{6x^2+x-15}{3x^2+11x+10}\,=\,0\)

6x² + x - 15   =   0

6x² - 9x + 10x - 15   =   0

3x(2x - 3) + 5(2x - 3)   =   0

(2x - 3)(3x + 5)   =   0

x  =  3/2   and   x = -5/3

f(g(x))  is undefined when  x = -2 ,  x = -5/3 ,  and  x = 3/2  .

-2  +  -5/3  +  3/2   =   -13/6

Thank you so much!

"y"  is supplemental to the  131° angle  =  49°

And since the angle at the bottom left of the small triangle containing "t"  is a vertical angle to "y", its measure  is also 49°

And the angle supplemental to the 142°  angle = 38°

So  "w"  in the small triangle on the bottom right =  180  - 49 - 38  = 93°

And the vertical angle to "w" in the small triangle at the top right  also  = 93°

So...the remaining angle in this triangle  =  180 - 93 - 42  = 45°

But this angle is a vertical angle to the angle at the bottom right in the small triangle at the top

So   ...  "t"     =    180  - 49  - 45  =  86°

Thank you so much!

AD = CD

AB  = CB

And DB  = DB

So...by  SSS, triangle DAB  is congruent to triangle DCB

But  this implies that  angle BDA  =  angle BDC

So...   BD bisects angle ADC

To prove the second part.....

Since  AD  = CD, then the angles opposite those sides in triangle ADC are also equal

So  angle DAC =  angle DCA

So

angle BDA  = angle BDC

angle DAC  =  angle DCA 

And  DE  = DE

So....by AAS, trianlgle   ADE  is congruent to triangle CDE

But this implies that  angle AED  =  angle CED

But...  m  AED  +  m CED  =  180

So, by substitution   m AED + m AED  = 180

2m AED  =  180       divide by 2 on both sides

m AED  = 90   =  m CED

And a line standing upon a line forming two right angles at their intersection means that the lines are perpendicular....so, .AC is perpendicular to DB 

What values for  c  make the equation  f(x) = c  have  6 solutions?

For example....

f( -5.5 )  =  6      and     f( 5.5 )  =  6

Therefore  f(x)  =  6  has  2  solutions.  They are  x = -5.5  and  x = 5.5 .

f(x)  =  -3   has  6  solutions.

f(x)  =  -4   has  6  solutions.

Those are the only integer values for  c  that make  f(x) = c  have  6  solutions.

-3  +  -4   =   -7

PM me and i will send it to you

Opposite sides of a parallelogram are congruent

So

NM  = OL

x +  15  = 3x + 5     subtract  x, 5  from both sides

10  =  2x       divide both sides by 2

5  = x

So....using  either equation     (5) + 15  =  20  =   NM  = OL   ⇒ 2nd answer

The diagonals of a parallelogram bisect each other

So this implies that 

DH  =  HF        and   GH  = HE

So

x + 3  =  3y      rearrange as    x  -  3y  =  - 3     (1)

4x - 5  = 2y + 3 rearrange as    4x - 2y  = 8  ⇒   2x  - y  =  4   (2)

Multiply (1)  by  -2  and we have that

-2x + 6y = 6          add this to  (2)  and we get that

5y  = 10   divide both sides by 2

y  = 2 

And using (1)

x - 3(2) =  - 3

x - 6  =  -3      add 6 to both sides

x =  3

So   {x, y}   =   {3, 2)  

Let us find the determinant of this system

1  2  0   1  2

2  0  1   2  0

3  2   1  3  2

( 0  + 6  + 0 )  - ( 0 + 2 + 4)

6  -  6  =  0

This indicates that we either have no solutions or infinite solutions

Gretzu, we could solve this with Gaussian elimination, but it seems easier to just use some Algebra

We  have

x  + 2y    = 2 ⇒  2y  =  2 - x    (1)   

2x  + z  =  1       ⇒  z  =  1 - 2x    (2)

3x + 2y + z  = 3      (3)

Sub (1) and (2)  into (3)   and we have that

3x + (2 - x)  + (1 - 2x)  = 3     simplify

3  =  3

This is always true....so we have infinite solutions

Holding x  constant  we have the solutions   {  x, (2 - x) / 2  , 1 - 2x }

In parallelograms, opposite angles are congruent.....so

WXY  is congruent to WZY

5. Let s and t be the solutions of the quadratic 4x^2 + 9x - 6 = 0. Find

s           +        t                         s^2  + t^2

__                ___       =             ________

t                     s                             st

This is much like 4 with a few twists

The sum of the roots  =     -9/4

So   s + t  =  -9/4

Square both sides

s^2 + 2st + t^2  =   81/16     (1)

The product of the roots  =   -6/4   =  -3/2

So  st  = -3/2   ⇒   2st  =  - 3    (2)

Sub  (2) into (1)  and we have that

s^2  - 3  + t^2  =  81/16         add 3 to both sides

s^2 + t^2  =   81/16 + 3

s^2 + t^2  =  81/16 +  48/16

s^2 + t^2  =  129/16

So

s^2 + t^2                     (129/16)              - (129/16) (2/3) =  - (129/3) (2/16)  =   

_________     =          _________  =    

   st                              - (3/2) 

- 43 ( 1/ 8)  =

-43 / 8