CPhill

Let D be the center of the red circle  and  r its radius

Let E be the center of the semi-circle with BC as a diameter

The radius of the large semi-circle = 2

We can form right triangle BDE with BD = 2-r , BE = 1 and the hypotenuse = ED = 1 + r

So

2^2  + (2-r)^2  = (1 + r)^2

4 + 4 - 4r + r^2  =  1 + 2r + r^2

7 = 6r

r = 7/6

sin A = 1/3 =  BD/ AB  = BD / 24  →  BD =  8

sin C =  1/2 =  BD / BC  = 8 / BC →  BC =  16

y =  x^2 -8x + (8 + c)

This is aparabols that turns upward

The x coordinate of the vertex  =    8 / (2*1)    =  4

And we want the associated y coordinate of the vertex to   = 1

So

1 = (4)^2 - 8(4) + (8 + c)

1  =  -16 + 8 + c

9  = c

(2x + 7) (x - 4) + 31 - jx  = 0

2x^2 - ( 1 + j) x + 3 =  0

Thiswill have one real solutionwhen the discriminant  = 0

(1 + j)^2  - 4(2)(3)  =  0

(1 + j)^2  =  24        take  both roots

1 + j =  sqrt 24           and     1 + j  =-sqrt 24

j =  sqrt (24)  - 1                    j =  -sqrt (24) - 1

j = 2sqrt (6) - 1                     j = -2sqrt (6)  -1

Let E be the center of the large semi- circle

Note that the radius of this semi-circle is  = 2

So AB = EA = EB  = 2

So triangle AEB is equilateral  and its area  =  (1/2) 2^2 * sqrt (3)/. 2 =  sqrt (3)     (A)

And the  area between AB and  the edge of  the larger semi-circle =  pi * 2^2 * 60 / 360 =  (2/3)pi     (B)

So....the  white area of the small semicircle = B - A =   (2/3)pi -sqrt (3)

And the radius of the small semi-circle = 1

So the area of the small  semi-circle  = (1/2) pi (1)^2  =  pi /2

Shaded area =  pi/2 - ( 2/3 pi -sqrt 3) = sqrt (3) - pi/6

(x + 1) (x-2p)  =

x^2 + (1 -2p)x -2p

This is a parabola that turns upward

The x value of the vertex =  - (1 -2p) / (2) =  -1/2 + p 

We want this x value to produce  0...so....

(x + 1) ( x -2p)  = 0

( -1/2 + p + 1 ) ( -1/2 + p -2p)  = 0

(1/2 + p) ( -1/2 -p)  = 0

(1/2)+ p = 0               -1/2 - p = 0

p = -1/2                      p =-1/2

p = -1/2

The line has a slope = 5

The slope at any point on the parabola =  -12x

Set these = 

-12x = 5

x  = -5/12

So y= -6(-5/12)^2 =  -25/24

So

-25/24  = 5 (-5/12) + c

25/12 - 25/24 = c

50/24 - 25/24  =   c = 25/24

https://web2.0calc.com/questions/find-all-pairs-x-y-10-of-real-numbers-such-that-x

https://web2.0calc.com/questions/coordinates_11172

\( \dfrac{1}{\sqrt2+sqrt3}+\dfrac{1}{\sqrt2-sqrt3}. \)

[ sqrt 2 - sqrt 3 + sqrt 2 + sqrt 3]

__________________________ =

[sqrt 2 + sqrt 3] [ sqrt 2 -sqrt 3]

     2sqrt 2

____________  =

     2     -   3 

- 2sqrt 2