CPhill

Simplfy as

6x^2 - 21x -31  = 0

Largest  x =   [ 21 + sqrt [ 21^2 - 4*6 * -31 ]  ] / [2 * 6] ≈  4.62

8x^2 + 6x - 1  =  0

ab  = -6/8 = -1/8

a + b  = -6/8 = -3/4          square both sides

a^2 + 2ab + b^2  = 9/16

a^2 + b^2   = 9/16  - 2ab

a^2 + b^2  = 9/16  +1/4

a^2 + b^2  =  13/16

a^3 + b^3  = 

(a + b) ( a^2 + b^2  -  ab)  =

(-3/4) (13/16 + 1/8)  = (-3/4)( 13/16 + 2/16)   (-3/4) ( 15/16)  = -45 / 64

No bounded region exists

https://www.desmos.com/calculator/uh6sw4gpdm

No prob.....I have those days when I should have just stayed in  bed, too....

S1 = 5

S2 = 15

Then the second term is 10   and  d = 5

15th term  is  5 + 14(5) =  75

S15  = [ 15th term + first term ] * no.of terms /  2 =

 [ 75 + 5 ] (15 / 2)   = 80 * 15/2  =  40 * 15  = 600

a1 + a3 =  12

a1  +  (a1 + 2d)  = 12

2a1 + 2d  =  12

a1 + d =  6

d = 6 - a1

a2 + a4 + a6  = 8

(a1 + d) + (a1 + 3d) + (a1 + 5d)  = 8

3a1 + 9d  =  8

3a1 + 9(6 - a1)  =  8

3a1 + 54 - 9a1  = 8

-6a1  = -46

a1 = -46 / -6  =  23 / 3

Simplify as 

x^2 - mx + 14 =  0

Roots             m

-1  -14            15

1    14           - 15

-2   -7               9

2     7              -9 

n + n + d + n + 2d  = 180

3n + 3d  =  180

n + d =  60

n = 60  - d

( n + 2d)  - n =  56

2d = 56

d = 28

n = 60 - 28 =  32°

sqrt [ 4x + 1]   not well defined whenever  x < -1/4           x ≥   -1/4

sqrt [ 8 -3x ]   not well defined when x > 8/3                     x ≤  8/3

sqrt [ 2x - 5 ]  in the  denominator  not well defined when x ≤  5/2              x > 5/2

The most restrictive interval is  (5/2 , 8/3 ]   { well-defined here }

Note....either  l a  l  =  30       or   l -a  l  =  30

x^2 - 30x - 1 =  30

x^2 -30x - 31  =  0

(x -31) ( x + 1)  = 0

x = 31   and x =  -1

Also

- (x^2 - 30x - 1)   = 30

-x^2 + 30x + 1 = 30

x^2 - 30x - 1 = -30

x^2 - 30x + 29 =  0

( x - 1) ( x -29)  = 0

x  = 1  and  x =  29