CPhill

(x^2  + 7/x)^4

There is no constant term

Note the expansion :

x^8 + 28 x^5 + 2401/x^4 + 294 x^2 + 1372/x

x^2 - l x^2 l

Note that the absolute value of  x^2  =  x^2

So......this always evaluates to   0

A slight mistake above

We know  that c = 4   and a  =1

So

-10  =  (1)^2  + b(1) + 4 

- 10  = 5 + b

-10 -5  = b   =  -  15

The form is

f(x)   =x^2  - 15x  +  4

We need to find the area of the table top

The radius of the table top can  be found as

4.71 / ( 2*3.14) =  .75 ft

The area =  3.14 (.75)^2   = 1.776 sq ft  = 1.8  sq ft

The sum of the angles = 180°

So

10x -22  + 4x  + 4x + 10   =  180   simplify

18x - 12  = 180

18x = 180 + 12

18x = 192 

Angle ABC  =  4 (32/3)  =  128/3 ≈ 42.66°

Angle ACB = 4 (32/3) + 10  ≈  52.66°

Angle CAB  = 10 (32/3) -22  =  320/3 - 22  ≈ 84.66° 

sin 2x  =  sin x      (using an identity on the left and simplifying)

2sinxcosx  - sin x  =  0            factor out sin x

sin x ( 2cos x  - 1)  =   0

We have two equations

sin x = 0    and this happens  at 0 , pi  and 2pi

2cos x - 1 =  0

2cos x  =  1

cos x =1/2    and this happens at pi/3  and  5pi/ 3

So the solutions are   0 , pi/3 , pi, 5pi/3  and 2 pi

f(x)  =  2x + 7  if x  < - 2                      f(x)  = -x^2 - x + 2  if x > = -2

-5 = 2x + 7                                           -5  = -x^2 - x + 2         rewrite as

-5 - 7  = 2x                                           x^2 + x - 5 - 2  =  0

-12 = 2x                                               x^2 + x - 7 =  0       complete the square on x

-12/2  = x  =  -6                                   x^2 + x  + 1/4  = 7 + 1/4

                                                            (x + 1/2)  =  29/4        take the positive root

                                                              x + 1/2  = [ sqrt (29) ] / 2

                                                              x  =  [ sqrt (29)  / 2  - 1/2  

The sum of the x values that make this true  are    -6 + [ sqrt (29) ] /2  - 1/2  =   -13/2  + [ sqrt (29)] / 2 ≈ -3.807

Call A the smaller and B the larger

A = 2* 8  =  16

B = 4 * 24  = 96

l A - B l   =  l 16 - 96 l  =   l -80 l  =   80

We have this relationship  (secant - tangent theorem )

PY^2  = AP * PB

11^2  =  15 * PB

121 / 15  = PB  

So

AB = AP - PB

AB  = 15 - 121 / 15  =   (225 -121) / 15   =   104 / 15

Note that triangles  ACD ,  CBD  are  similar  and each are 30-60-90 right triangles

And since  AD is opposite a 60° angle it =  sqrt 3 * CD =  sqrt 3 * 8 sqrt 3 =  24

And, by similar triangles, we have the following relationship

DB / DC  = DC / AD

DC^2 = DB * AD

(8sqrt 3)^2  = DB * 24

256 / 24  = DB  = 32 / 3  (cm)