CPhill

Simplify

(7 + n) x  <  -36

x  <  -36  / ( 7 +  n)

n cannot =  -7

3. SOP and SAP  are insribed angles intercepting the  same  arc....so, they are equal

So

4 SOP - 174  =  SOP

3SOP  = 174

SOP  = 174 / 3  =  58  =  SAP

4.  Since MR  = MK, then arc  MR =  130

Since arc MQ = 28

So arc RQ =  arc MR - arc MQ   =  130 - 28  = 102

Then angle QKM  = 14

Angle KPM =  (1/2) (arc RQ + KM)   = (1/2) ( 102 + 130)  = 232  / 2  =  116

Then angle RPK =  180 - angle KPM   = 180   -  116    =  64

1.   Angle ABE  =  Angle FBC   = (35 + 80) /  2  =  57.5

Angle ABC  = 180  - 57.5  = 122.5

2. Arc EC + DB =  360 - 170  - 110  =  80

So EC + DB = 80   (1)

Then 

30 =  (1/2)(EC -DB)

60  = EC - DB    (2)

Using (1) , (2)

EC + DB =  80

EC - DB =  60           add these

2EC  = 140

EC =  70

Simplify as

5x =  2y^2 - 6y + 13

5x =  2 ( y^2  - 3y)  + 13        complete the square on y

5x  = 2 ( y^2 - 3y + 9/4)  + 13 - 18/4

5x = 2 ( y - 3/2)^2  + 17/2             divide through by 5

x = (2/5)(y -3/2)^2 + 17/10

The vertex is ( 17/10 , 3/2)  =  (1.7, 1.5 )

Simplify as

x^2 - 2x + y^2 + 6y  = 26            complete the square on x,y

x^2 -2x + 1  + y^2 + 6y + 9 =  26 + 1 + 9

(x - 1)^2 + ( y + 3)^2   = 36

This is a circle  centered at (1, -3)  with a radius of sqrt (36)  = 6

Area = pi * 6^2  =   36 pi

Here :   https://web2.0calc.com/questions/square_32

Simplify as

x^2 -2x + 21             complete the square on  x

x^2  -2x + 1 + 21  -  1   

(x - 1)^2 + 20    =

1 ( x+  - 1)^2 + 20

\(\frac{x+4}{x+5} = \frac{x-3}{2x + 8} - 7 \)

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

(x + 4) / (x + 5)  =  -[ 13x + 59  ] / (2x + 8)

(x + 4) / ( x + 5)  + [ 13x + 59 ] / (2x + 8)  =  0

(x + 4) (2x + 8) + (x + 5)(13x + 59)  = 0

2x^2 + 16x + 32  +  13x^2 + 124x + 295  = 0

15x^2 + 140x + 327  =   0

x =  [ -140  + sqrt [ 140^2 - 60*327 ] ]/ 30  = 

[ -140 + sqrt (-20) ] / 30   =

[ -70 + i sqrt (5) ] / 15

And

x =  [ -70  - i sqrt (5) ] / 15

19x^2  - 6x + 31   =   0

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

[ 6 + sqrt [ -2320] ] /  38  =

[6 + i [-16 * 145 ] / 38   = 

[ 6 + 4i sqrt [145] ] / 38  = 

[ 3 + 2sqrt [ 145 ] i ] / 19  =

3 /19   +  (2/19)sqrt (145)  i

a =  3/19       b = (2/19) sqrt (145)

ab =   (3/19) (2/19) sqrt (145)  =  

(6 / 361) sqrt (145)

(15 - 16i) /2  - 3i (2) / 2   =

[ 15 - 16i - 6i ]   / 2  =

15/2  - (16i -6i ) / 2  =

15/2  - (22i)  / 2  = 

15/2 - 11i