CPhill

31 * 32¹  +  8 * 32⁰   = 1000  =  v8

25₆  =  2*6 + 5  =   17  base 10

54₆ =   5*6 + 4  =    34 base 10

17 + 34  =  51  base 10

To convert back to base 6

51  /  6    =   8 R  3

8 / 6  =         1 R   2

1/6  =            0 R 1

123₆  

Convert 12121₃  to base 10

1 * 3^4  + 2 * 3^3  + 1 * 3^2 + 2 * 3  + 1    = 151

Convert 1222₃ to base 10

1* 3^3  + 2 * 3^2 + 2 * 3^1  + 2   = 53

151  + 53   =  204

Convert  204₁₀  to base 3

204 / 3  =    68  R 0

68 / 3  =    22  R 2

22 / 3  =  7 R 1

7 / 3  =    2 R 1

2 / 3  =    0 R 2

So

12121₃ + 1221₃  = 21120₃

(x + 1) ( x -2p)

x^2 + ( 1 - 2p)x - 2p

The x cootdinate of the  vertex is  - [ 1 -2p ] / 2  = [ 2p - 1]  / 2

We   want   this point on the graph  ( [ 2p -1 ] /2 , 0 )

So

( [2p-1]/2 )^2 + (1 - 2p) ([2p-1] /2) -2p =  0

p^2 - p + 1/4  + p - 2p^2 -1/2 + p - 2p =  0

-p^2 - p -1/4 =  0

p^2 + p + 1/4  = 0

(p + 1/2)^2  = 0

p + 1/2   = 0

p  = -1/2

If your team  has one win and 11 losses after four weeks....and  then wins one  game and loses two per week after that, you can't ever get to 60% !!!

Since the ratio between succesive terms is the same we  have

[100 +n ] / [ 60 + n]  = [ 140 + n ] / [ 100 + n ]     cross- multiply

[100 + n ] [ 100 + n ]  = [ 60 + n ] [140 + n]

10000 + 200n + n^2  =  8400 + 200n + n^2

10000   = 8400 

No solutions exist

a1 + a3  = 5

a2  + a4 = 6

a2  = a1 + d

a3 = a1 + 2d

a4 = a1 + 3d     where d = the difference  in  successive terms....so....

a1 + (a1 + d)  = 5

(a1 + d) + (a1 + 3d)  = 6         simplify these

2a1 + d = 5         (1)

2a1 + 4d = 6       (2)

Subtract (1) from (2)

3d  = 1

d =1/3

Using (1)

2a1 + (1/3)  =  5

2a1  =  14/3

a1  =  14 / 6  = 7 / 3

https://web2.0calc.com/questions/algebra_40599

range  = [ -7 , 11 ]

obviously....x cannot  = 0

1 + 3/x   cannot = 0   ...so   1 + 3/x  =  [ x + 3] / x

And    2/ ( [ x + 3] / x)  =  2x / [ x + 3]

So x cannot =  -3

Lastly

1 + [2x] / [ x + 3]   cannot  = 0

[ x + 3 + 2x] /[x + 3]

[ 3x + 3 ] / [x + 3]

So......  1 / [ (3x + 3) / (x + 3) ]

(x + 3 )  / (3x + 3)      

So x cannot = -1/3