CPhill

x^3 ( x^2 + tx)  =  x^5 + tx^4.....it doesn't have an x^2 term anyway

x^2 + 23x  + k

To have a double root, the discriminant  must =  0 

23^2  - 4 *1 * k   =  0

529 =  4k

k = 529 / 4

16r^3 = 26

r^3  = 26/16  = 13/8

r = (13/8)^(1/3)

24th term =  16 (13/8)^(1/3)  ≈  18.81

x^2 + 8x +  p

p  = (1/2 the coefficient on  x )^2  =  (8/2)^2  =  4^2 =  16

So

x^2 + 8x + 16  =

(x + 4) ( x + 4)  =

(x + 4)^2

We want the form of the ellipse to be    x^2 / a^2  + y^2 / b^2  = 1

x^2 + 2y^2  =  32       divide through by 32

x^2 / 32  +  y^2 / 16  =   1

b^2   = 16

b   =  4 =  semi-minor axis   =   the   radius of the  circle....so r^2 =  16

x^2 + y^2  = 16 

Simplify as

19x^2- 6x + 31  =  0

x = [   6 +/- sqrt [ 36 - 4*19*31] ] / 38  =  [ 6 +/- sqrt [-2320] ]/ 38  = 

[ 6 +/- 4 i sqrt [ 145] ]  / 38

a = 6/38  = 3/19

b = 4sqrt [145 ] / 38  =  2sqrt [ 145] /19

a * b  =    (3/19)  (2sqrt [145] / 19 )  =  6sqrt[145] / 361

Multiply through by  (x + 5) (2x + 8)

(x + 4)(2x + 8)  = (x - 3)(x + 5)  - 7(x+5)(2x + 8)

2x^2 + 16x + 32  =  x^2 + 2x -15  - 7 (2x^2 + 18x + 40)

2x^2 + 16x + 32 = x^2 + 2x - 15 - 14x^2 - 126x - 280

15x^2 + 140x + 327  =  0

The solutions  are

[-140  +/- sqrt [ 140^2 - 4*15*327] ] / 30    =   [-140 +/- sqrt [ -20 ] ] / 30  =

[ -140 +/- 2isqrt (5) ] / 30   =

-14/3  +/- (sqrt (5)  / 15)  i

Simplfy  as

-8x^2 -kx + 47   = 0

We have no real solutions when the discriminant is <  0

k^2 - 4 (47)(-8)  <  0

k^2 + 1504 < 0

The left side is always  positive for any k.....so.......we will have real solutions for any k

80  = (1/2) (12) ( b + b + 4)

80 / 6  =   2b + 4

40/3  = 2b + 4         multiply through by 1/2

20/3  = b + 2

b = 20/3 - 2   = 20/3 - 6/3 =    14/3

b + 4 =  14/3 + 12/3  =   26/3

Assuming that we have an  isosceles trapezoid, we  can  find the length of a side by constructing a right triangle with legs of   12  and  [ (b + 4) - b ] / 2  = 2

The side length is  sqrt [ 12^2 + 2^2]  = sqrt (148) = 2sqrt (37)

The perimeter  is    14/3 + 26/3 + 2 * 2sqrt (37)  =  40/3 + 4sqrt (37)  ≈  37.66

Simplify as

x + y + 3z   =10

6x - 6y - 13z = -7

( k-1)x - 2y + z = 3

Multiply the first equation by  6....add to the second equation

6x + 6y + 18z   = 60

6x - 6y  -13z  =  -7       

12x + 5z  = 53    (1)

Multiply the first equation by 2 ....add to the second equation

2x + 2y + 6z  = 20

(k-1)x - 2y + z  = 3         

( 1 + k)x + 7z  = 23      ( 2)

Multiply (1) by 7    and (2)   by 5

84x + 35z  = 371

5(1 + k)x  +  35z = -115

This will have no solutions when

5 (1 +k) =  84

5 + 5k = 84

5k = 79

k = 79/5