CPhill

3x^2 -5x -6 = 0

product of roots

ab = -6/3  =  -2

(ab)^2   = 4

2ab = -4

sum of  roots

a + b  =  5/3          square both sides

a^2 + 2ab + b^2  =  25/9 

a^2 - 4 + b^2  = 25/9

a^2 + b^2  = 25/9 + 4

a^2 + b^2  =  61/9        square both sides again

a^4 + 2 (ab)^2  + b^4  =  3721 / 81

a^4 + 2 (4) + b^4  = 3721/81

a^4 + b^4  + 8 = 3721/81

a^4 + b^4 = 3721/81 - 8  

a^4 + b^4  =  3073 /  81

Slope  =  [ 2  - - 7 ]  / [ -8 -4 ]  =   9  /  -12   =  - 3/4

Using (4, -7)

y = - (3/4) ( x -4)  - 7                    multiply through by 4

4y  = -3 ( x -4) - 28

4y  =  -3x + 12   - 28

3x + 4y  =   - 16

        P                    Q

                    4

                   X

                   8

   S                               R

PXQ  and RXS are similar triangles

The scale factor from  RXS and PXQ =  sqrt (8/4)  = sqrt (2)    /  1

Thus the height  of RXS =  sqrt (2) h / ( 1 + sqrt 2)              where h is the height of the trapezoid

And the height of PXQ  =  h /( (1 + sqrt 2) 

And  the base of PXQ =   base of RXS/sqrt 2

Using PXQ

Area =  (1/2) (base of RXS)  /sqrt (2) * h / ( 1 + sqrt 2)

4 = (1/2) (base RXS) * h / ( 2 +sqrt 2)

8(2 + sqrt 2) /  base RXS  =  h

Area of trapezoid =  (1/2) h (sum of bases)

Area of trapezoid =  (1/2)[ 8 ( 2 + sqrt 2) ] / base of RXS ]  [ base of RXS + base PXQ   ]  =

 4 (2 + sqrt 2)  / [ base of RXS ]  * [ base of RXS * ( 1 + 1/ sqrt 2) ]  = 

 4 [ 2 + sqrt 2 ] [ 2 + sqrt 2 ]  / / 2    =

2 [ 2 + sqrt 2] ^2  =

2 [ 4 + 4sqrt 2 + 2] = 

2 [ 6 + 4sqrt 2]  =

12 + 8sqrt 2

      X                Y

90

W60                          30  Z

Triangle ZXW  is a 30 - 60 -90  right triangle

Angle XZW  = 30°

3x - 6y  = 12 + 8x -3y            simplify

-3y = 5x + 12                      divide through by  -3      

y = (-5/3)x - 4

y  intercercept = (0, -4)

x intercept

0 = (-5/3)x  -4

4 = (-5/3)x

x = -12/5 = -2.4

Distance between   intercepts  =  sqrt [ (-2.4)^2 + (-4)^2 ] =  sqrt [ 544/25 ] =  (4 sqrt 34  )  /  5

Points are  (0,14)  and (x , 0)

slope =  ( 14 - 0 )  / ( 0 - x)  = 8

14 / -x = 8

-x = 14/8  = -7/4   = x intercept

Radius of large circle =  9......area =  pi * 9^2  =  81 pi cm^2

Total area of smaller circles=  (7) pi (3)^2  = 63 pi cm^2

Total area of a's  =  81 pi - 63pi  =   18pi cm^2

Using the square of the  distance  formula

(a-2)^2  + (2a - 7  - -1)^2 =  5

(a -2)^2  +  (2a -6)^2  = 5

a^2 -4a + 4 + 4a^2 -24a + 36  =  5

5a^2 - 28a + 35  =   0

Sum of possile a's  =  28  / 5

AE = AB /2  = 4  = BE

CE = 2

Intersecting Chords Theorem

AE * BE = CE * ED

4 * 4 =  2 * ED

16  = 2 * ED

ED  = 16 / 2    = 8

AD^2  = AB ( AB + BC)

AD^2  = AB ( AC)

4^2   =  AB (12)

16  = 12 AB

16/12  = AB  =  4/3