CPhill

x^2 + 6x - 91  = 4x - 70

x^2 + 2x - 21  =  0

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

x^2 + 2x + 1  = 21 + 1

(x + 1)^2  =  22        take the  positive  root

x + 1  = sqrt (22)

x = sqrt (22)  - 1                (this is the only positive value of x that solves the equation)

Factor the polynomial in the  numerator  as  (c - 3) ( c + 9)

This gives us

(c+ 9)  + 2c =  23

3c + 9  = 23

3c =  14

c = 14 / 3

If x = 2 , then f(x)  =  0

If x = -1, then f(x)  =  1

The domain is  [ -1 , 2 ] 

See the following

Angle BCE  = 90    = Angle ADE

Because BC and DA are parallel, then angles EAD and EBC are opposite interior angles between these  parallels.....therefore they are equal

So by AA congruency, triangles EAD and EBC are similar

And  AD = 2 and BC = 1......therefore the ratio of the  sides = 2 : 1

Then BE =  [ 1 / (1 + 2) ] (AB)  =  ( 1/3)(5)  = 5/3

And by the Pythagoren Theorem, CE = sqrt ( BE^2 - BC^2) =  sqrt (5/3^2 - 1^2)  = sqrt (16/ 9) = 4/3

And AE =  5 - 5/3 =  10/3

And by the P Theorem again , ED = sqrt (  AE^2 - DA^2) =  sqrt [ (10/3)^2 - 2^2] =  sqrt [ 64/9] = 8/3

So CD =  CE + ED =  4/3 + 8/3 =  12/3 =  4

And ACBD is a parallelogram  with bases of 1,2   and  a height of 4

So [ ACBD ]  =  (1/2) (height) ( sum of the bases)  = (1/2) (4) ( 1 +2)  = 6 

Very nice, Bosco ....

Your reasoning  and   presentation  are  very good  !!!

Equation of first circle

(x - 2)^2 + ( y + 1)^2 =  16   → x^2 - 4x + 4 + y^2 + 2y + 1  =  16

Equation of second  circle

(x + 2)^2  + (y - 5)^2 = 100 → x^2 + 4x + 4 + y^2 - 10y + 25 = 100

Subtract the second equation  from the  first

-8x + 12y -24 = -84

-8x + 12y  = - 60

-2x + 3y = -15

3y = 2x - 15

y=   [  (2/3)x - 5 ]     sub this into the first equation  for  y

(x - 2)^2  + [ (2/3)x - 5 + 1]^2  = 16

(x - 2)^2  + [ (2/3)x - 4]^2  = 16

x^2 - 4x + 4 + (4/9)x^2 - (16/3)x + 16 = 16

(13/9)x^2  -(28/3)x + 4  = 0

13x^2 - 84x + 36 = 0 

(13x -6) ( x - 6) = 0

Setting both factors to 0  and solving  for  x we have that

x = 6/13     and y = (2/3)(6/13) - 5  =  12/39 - 5  =  -61/13

x= 6    and y=  (2/3)(6) - 5 =  -1

(AB)^2 =  ( 6 - 6/13)^2 + ( -61/13 + 1)^2  =  (72/13)^2  + ( -48/13)^2  =  576 / 13

3x^2 + nx  + 72

n will be greatest when 72 is  factored as  72 * 1

Factor  as    (3x  +1) (x + 72)   →    n =  217 x  

x= 12  y = 6

xy = 72

The error occurs in the  second step

(x+ 4) ( 2x +1) (x+2)(x - 2) = 0

(2x^2 + 9x + 4) (x^2 - 4) = 0

2x^4 + 9x^3 + 4x^2 - 8x^2 - 36x -16 = 0

2x^4 + 9x^3 - 4x^2 -36x - 16  = 0

Here is a representation :

Notice that angles AEC  and BED are equal (vertical angles)

And angle ACE = angle BDE are also equal (right angles)

Then by AA congruency, triangle BED   is similar to  triangle AEC

And  BD = 6  and AC = 2

So  the ratio of the  sides =  3 : 1

Then BE = (3 / (3 + 1)  * AB  =  (3/4) (10) = 7.5

So AE = 10 - 7.5 = 2.5

Then using the Pythagorean Theorem twice,  CD =  sqrt [ 7.5^2 - 6^2 ] + sqrt [ 2.5^2 - 2^2 ] =

sqrt [ 20.25] + sqrt [ 2.25] =

       4.5    +  1.5  =

              6