CPhill

Sum of first ten terms  -  Sum of first nine terms =   tenth term  ......so....

(10)(10+1)(10-2)  -  (9) (9 + 1) (9 - 2)   =

880  -    630   =  

250  = tenth term

Rearrange as

-11x + y =   14

  ax + y  =   - 4   

This system will have no solutions if  a  =  -11

1/x  + 1/y =  3

1/x -  1/y =   7                   add these equations

2/x  = 10           

2/10  = x   =  1/5

So

1 / (1/5)  + 1/y  =  3

5   +  1/y   =  3

1/y  =  3 - 5

1/y  =  -2

-1/2  =  y

So

x +  y  =  

1/5    -  1/2   = 

2/10  -  5/10  =

-3 / 10

-x + 3 =  0                          2x  + 1  =  0

3 = x                                   2x  = -1

                                            x = -1/2

3 +  -1/2 =    5/2

Third term =  2^3 + 8^3 =  520

Fourth term =  5^3 + 2^3 + 0^3   =  133

Fifth term =  1^3 + 3^3 + 3^3 =  55

Note an interesting thing

Sixth term  =  5^3 + 5^3    =  250

Seventh term =  2^3 + 5^3 + 0^3  =  133

So.....starting with the fourth term , the series repeats in a  cycle of three terms

Area of triangle DPE  can be found as

Semiperimemter, S,   =   [2.5 + 2 + 1.5 ]  / 2  =  3

Area  =  sqrt [  3 ( 3 - 2) (3 - 2.5) ( 3 -1.5) ] =   1.5  

By the property  of  medians

PC =  2 * PE  =  2 * 1.5  =  3

PA =  2 *  PD =  2 * 2  =  4

And triangles  DPE and APC are similar  with each side of APC twice as long as that of DPC

So scale factor of triangle   APC   to  triangle DPE   =   2

Area of triangle  APC =  area of DPE * scale factor^2  =   1.5 * 2^2  =  6

And by another median property areas of triangle  AEP  plus triangle  CDP   = area of triangle APC

So....area of  AEDC  =   1.5 + 6  + 6   =     13.5

Any x that makes the denominator =  0    is not in the domain

So

x = 9    and x =  -9  are not in the domain

Domain  = 

(-inf , -9) U (-9, 9)  U  (9 , inf)

See the image here

Area of ABC  = 

[DFEC ] -  [DBA] - [AEC]  - [BFC]  =

[6*9 ] -  [ (1/2)(4)(3) ]  - [(1/2) (6) (6) ] - [(1/2) (2) (9) ] = 

54  - 6    - 18  -  9   =

21

If x  = 3  then

3(3)^2  + 2   =   29

So  we need to have

a(3)^2 + 2  =  29

9a  = 27

a = 3

See the graph here : https://www.desmos.com/calculator/srad0t3mxn

3x^2  + 4x  +  1  =  a( x + h)^2 + k

3x^2 + 4x + 1  =  a (x^2 + 2xh + h^2) + k

3x^2 + 4x + 1  =  ax^2 + 2ahx + ah^2  + k

a = 3

4  = 2ah

4 = 2(3)h

4 = 6h

h = 4/6 = 2/3

ah^2 + k =  1

3(2/3)^2 + k  = 1

(4/3) + k =  1

k =  1  - 4/3

k = -1/3