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1/j + 1/k =  2/3

( j + k)   / jk  = 2/3

j + k   =  (2/3)jk

3 ( j + k)  =  2jk

3j + 3k =  2jk

3k =  2jk - 3j

3k  = j ( 2k - 3)

j   =   3k  / ( 2k - 3)

 k    j

2    6

3    4

6    2

Sum ok all possible values  of  k =   11

Is there a diagram here   ???

3/5+4/5i

 ( -inf < x ≤  -  4 ]  U   [  3 ≤ x < 5 )

sqrt2

(-inf, -4] U [3,5)

If  1 +  sqrt 2  is a root  so is  1  - sqrt 2

So

(  x - (1 + sqrt 2))  ( x  - (1 - sqrt 2))   =   x^2  - 2x - 1

                              x  +  2

x^2   - 2x    -  1   [  x^3    +  0x^2  +   bx  +   c   ]

                              x^3    -2 x^2    -   1x

                            _________________________

                                        2x^2    + (b + 1)x   +  c

                                        2x^2    -    4x         -2

                                     ________________________

                                                    (b + 1 + 4)x + c + 2

                                                      (5 + b) x  + (c + 2)  =  0

This will   =  0   when    b =  -5     and  c=  -2

And  the interger  root  is

x + 2 =  0

x  =   -2

11. There are 4^2 + 4 + 1 = 21 functions.

12.The probability is 4/15 ==> 4 + 15 = 19.

13. The probabiilty is 65/231 ==> 65 + 231 = 296

14. The mean value is 28/5 ==> 28 + 5 = 33.

Take 125    multiply by 4 =  500.....take the positive and negative  roots of this

a =  ±sqrt ( 500)  = ± sqrt (5) * sqrt (100)  =  ±10sqrt (5)

11x^2  + 44x   - 99   = 0           divide through  by  11

x^2  + 4x   -  9    = 0               add 9 to  both sides

x^2 + 4x    = 9                       take 1/2 of 4 = 2.....square this =  4   add to  both sides

x^2 + 4x  + 4   =  9  + 4       factor the  left side,  simplify the right

(x + 2)^2  =  13

r = 2      s  =  13

r +  s  =    15

No figure is showing. 

There's a similar question here:   https://web2.0calc.com/questions/how-many-rectangles-are-in-this-figure-each-angle

_.

Morris ordered 66 trophies.

(b) There are 108/3 = 36 unique sets.

(c) For the first card, there are 3 ways to choose the number.  For the second, there 2 choices, and then there is only 1 choice.  So there are 3*2*1 = 6 ways that the numbers can be chosen.  Doing this for the other attributes, we get 6*6*24*6 ways.  But the order of cards doesn't matter in a set, so we divide by 3!: 6*6*24*6/3! = 864.  So there are 864 sets for part (c).

(d) The cards can have the same number, color, shape, or shading.  If all the colors are the same, then there are 6*24*6/3! = 144 sets.  If all the numbers are the same, then there are 144 sets.  We get the same number for shape and shading, so there are 4*144 = 576 sets for part (d).

(e) We need to choose two of the attributes.  There are C(4,2) = 6 ways of choosing two attributes.  For each of these two attributes, there are 3 options.  For the other two attributes, there are 3 ways of assigning the choices, so there are 6*3*3*3*3 = 486 sets for part (e)

(f) First we choose which attributes are the same.  There are C(4,3) = 4 ways of choosing which attributes are the same.  There are then 3*3*4 = 36 ways to assign which is which for each of these three attributes, and there are 4 ways to assign the choices for the fourth attribute, so there are 4*36*4 = 576 sets for part (f).

4/11 of the entire group passed the test.

s (1.085) = final cost after taxes

The zeroes of y = x^2 - 18x - 33  using the quadratic formula,   are x = 9 +- sqrt 114

   the POSITIVE root shows where the bowl shaped parabola becomes POSITIVE    9 + sqrt 114 = 19.67     ~~ 20 units will start showing a profit

The product of my numbers is 3*9 = 27.

IF it's:  y = 2/(x+1) 

x = 2 / (y+1)

y+1 = 2/x

y = 2/x    -1     is the f^-1

y = 2/(1/5) -1 = 9

First you'll have to find the inverse function

y = 2/ (x+1)      or is it     2/x    +1 ?      people often forget parentheses which are necessary.....

x = 3/4 and x = 1/2.

x² -7x - 30 = 0   Use Quadratic Formula    a = 1   b =  -7   c = -30

\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)         you will find    x = 10 and -3

(x-10)(x+3) = x^2 -7x-30     check !

y = k/ (sqrtx)       k = proportionality constant

15 = k/sqrt(24)

15 sqrt24  = k

Now when y = 3

3 = 15 sqrt(24)/x

x = 15 sqrt (24)  / 3  = 5 sqrt 24 = 10 sqrt 6   

The straight line would be 5^2 - 1^2 = 24, but then you have to sqrt it which gives √24, and the ? mark side is 5^2 + √24^2 = 49, and then sqrt, which gives 7