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a  =  x  +  y        (1) 

b  = x^2  + y^2   (2)  

c = x^3 + y^3        =  ( x + y)  (x^2  - xy   + y^2)   =  ( x + y) (x^2 + y^2 - xy )   (3)

Square both sides of   (1)   and we get that

a^2  =  x^2  + 2xy  + y^2

-2xy  = x^2 + y^2   - a^2

-2xy  =  b - a^2

-xy  =  [ b - a^2 ] / 2      (4)

Sub  (1), (2)  (4)   into  (3)    and we have that

c =  ( a)  ( b  +  [ b - a^2] / 2 )

c = (a) ( b + b/2  - a^2/2 )

c  = (a) ( (3/2)b - (1/2)a^2  )

c =  (3/2)ab  - (1/2)a^3

wouldn't it be 7 chooose 4 instead of 8 chooose 5?

Thank you!

Good job....all correct!!!

We can only add/subtract  matrices of the same dimensions....

We have  two    sets  to  consider

(right  right   up    up   up  )    *   ( right right  right  right  up  up  up  )

We have   C(5,2)   ways to select  the positions for  "right"  in the first set  and  C (7,4)  ways to select the positions for "right"  in the second set

So....the total paths  are    C(5,2)  * C(7,.4)  =   10  * 35  = 350 possible  paths

[ Note....the same result  would be obtained  by choosing any C(5,3) and C(7,3)  positions for  "up"  in both sets ]

EDIT....THX TO THE GUESY FOR CATCHING MY ERROR  !!!!

Center  =  (-8, 4) =  (h, k)

Since  the transverse  axis is parallel to the x axis, this hyperbola  opens left/right

The length of the transverse axis  =  4 units... so  a = 2

And   b =  12/2  = 6

So....the equation  is

(  x + h)^2           (y - k)^2

_________   -    ________   =  1

     a^2                   b^2

( x  + 8)^2               ( y  - 4)^2

___________   -    _________   =     1

        4                         36

Here's the graph  :  https://www.desmos.com/calculator/zuuet2usbm

[ 9              This goes in the first box

  7 ]

The inverse  is

   1                        [ 2   - 1      =                1            [ 2    - 1      This goes in the second box

___________       -3  - 4 ]                 -    ______     -3    -4]

(2)(-4) -  (1)(3)                                          11   

-1       [  2  - 1           [ 9        =         -1            [ 11

__         -3  -4 ]           7 ]            _______        -55]         This goes in the third box

11                                                   11

[ -1

  5 ]        This goes in the last box

The points   ( 1, 2)  ,  ( 4, 3)   and   (7, 4)   are on the graph  of f(x)

The inverse reverses these....so.....the points   ( 2, 1) , ( 3, 4)  and  (4, 7)   are on the inverse

f^-1(3)  =   4

So

f^-1  ( f^-1 (3)) =  f^-1(4)   =    7

Thanks guest, I have fixed mine now.  It was just a silly error on my part.

We are now in agreement.

Thanks! I understand the solution, too, now! 

Heureka had an ingenious solution,  here  : 

https://web2.0calc.com/questions/let-denote-the-circular-region-bounded-by-x-2-y-2

x^3  + 2x^2  - 5x - 6   =

Write this as

[ x^3  + x^2 ]  + [ x^2  -  5x  - 6  ]  =

x^2 ( x + 1) +  (x + 1) ( x - 6)  =

(x + 1)  ( x^2  + x  - 6)  =

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

-1  [ 1     -1     -   2     5   - 6   ]

               -1       2      0    -5

      ______________________

       1    - 2       0       5   -11

-2   [ 1   - 1      -2     5      - 6  ]

              -2       6    -8        6

       _____________________

       1      -3      4    -3        0

-3  [   1    -1     -2     5      -6  ]

                -3   12    -30   75

      ______________________

         1     -4    10    -25  69

We get alternating signs  on the bottom row......thus    -3   is the  lower  bound

See the graph, here : https://www.desmos.com/calculator/sxca8cg7bf

The graph of the function crosses through the x-axis at 2 and behaves in a cubic fashion. The graph of the function touches and bounces off of the x-axis at −5 and crosses straight through the x-axis at 0.

Given A + B = 225 = B = 225 - A= B = 180 + 45 - A = B = 180 + (45 - A) (As 225 = 180 + 45)Putting value of B in question we getcot A/(1 + cot A) * cot {180 + ( 45 - A)} / [ 1 + cot {180 + (45 - A)}= cot A/(1 + cot A) * cot (45 - A) / [1 + cot (45 - A)] [As cot (180 + A) = cot A]= cot A/(1 + cot A) * [ {cot 45cot A + 1 / cot A - cot 45} / {1 + (cot 45 cot A + 1) / (cot A - cot 45)}][As cot (A - B) = (cot B cot A + 1) / (cot B - cot A) ]= cot A/(1 + cot A) * [ { cot A + 1 / cot A - 1} / {1 + ( cot A + 1) / (cot A - 1)}] (As cot 45 = 1)= cot A/(1 + cot A) * [ { cot A + 1 / cot A - 1} / { (cot A - 1+ cot A + 1) / (cot A - 1)}] [ Taking L.C.M.]= cot A/(1 + cot A) * [{cot A + 1 / cot A - 1} / { (2 cot A) / (cotA - 1)}]= cot A/(1 + cot A) * (cot A + 1) / (cot A - 1) * (cot A - 1) / (2 cot A)= 1/2

So you are trying to find the reciprocal of that which is just 2.

To convert  .07 (repeating) to  a fraction

Let   x  = .07  (repeating)

Multiply both sides   by  100

100x  =  7.07 (repeating)

100x  - x     =   7.07(repeating)  -  .07(repeating)

99x  =  7

x  = 7/99

So

.07 -  .07 (repeating)  =

7/100  -  7/99  =

7(99) - 7(100)

___________   = 

100 *  99

7 (99 - 100)

__________   = 

     9900

7 (-1)

______   =

  9900

   -7 

_____

 9900

Your welcome, uniformly at random means choosing random ... 

This is just 7/99 - 7/100, or 7/9900.

~~GF

See here  :    https://web2.0calc.com/questions/please-help-i-really-don-t-know

im sorry i meant to post this as a screenshot

I will make up just 1 for you !!

5x^2  -  2x  + 60 = 2020

The length of the  major  axis  =   2a  =  420

So   a  =   210

The length of the minor axis  = 2b =  180

So   b  =  90

And   c  =     sqrt  ( 210^2  -90^2 )  = sqrt (36000) ≈  189.736 ft

And the distance  between Angela  and Zach  =  2c   =  2 ( 189.736)  ≈  379.4 ft 

a=1;b=0;c=0;p=0; cycle:d=a*100+b*10+c;if( b==1or b==1 and c==1 , goto loop, goto next); loop:printd,", ",;p=p+1; next:c++;if(c<10, goto cycle, 0);c=0;b++;if(b<10, goto cycle, 0);b=0;c=0;a++;if(a<10, goto cycle,0);print"Total = ",p

OUTPUT:
1 - There are 18 2-digit "memorable" integers.
2 - There are 99 3-digit "memorable" integers.
(10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51, 61, 71, 81, 91, 101, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 131, 141, 151, 161, 171, 181, 191, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919) = 117 such numbers.

See here :  https://web2.0calc.com/questions/system-of-equations_11

Call the side  of the  largest green triangle, S

Then the side  of the  largest  orange triangle  = (√3/2) * S

And the side  of  the second largest green triangle  =  (√3/4) * S

And the side  of the next largest orange triangle is    (3/8) * S

And the side  of the  last green triangle is   (3/16) * S

And the side of the last orange triangle  =   (3/32)√3 * S

So....the ratio  of the area  of the  largest  green triangle     to the  smallest orange  triangle  = 

S^2                                            1                   32^2          1024

______________  =          __________  =   _____   =    _____

 [ (3/32)√3* S] ^2                   27/ 32^2             27               27