CPhill

Simplify as

x^2 + 6x + 5  = 0 

(x + 5) ( x + 1)  = 0

x = -5    x  = -1

Let a  = -5   b = -1

a^3/b^3  + b^3 / a^3  =

(-5)^3 / ( -1)^3   + (-1)^3 / (-5)^3  =

125/1 + 1/125  =

125.008

Simplify as

8x^2 + 6x - 1

ab = -1/8

2ab = (-1/4)

a + b =  -6/8  = -3/4      square both sides

a^2 + b^2 + 2ab = -9/16

a^2 + b^2 -1/4  = -9/16

a^2 + b^2  =  -9/16 + 1/4  =   -5/16

a^3 + b^3 =  (a + b) ( a^2 + b^2  - ab)  = 

(-3/4) ( -5/16 - (-1/8))  = 

(-3/4) ( -5/16 + 1/8)  = 

(-3/4) ( -3/16)  = 

9 / 64

Reflection over  y = x  reverses the coordinates

(a,b)  → (b, a)

So

B = ( 7, 5)

Simplify as

7x^2 + 2x + 21  =  0 

uv =  21 / 7   = 3

(uv)^2   = 9

u + v = -2/7      square both sides

u^2 + 2uv + v^2  = 4/49

u^2  + v^2  + 2(3)  =4/49

u^2 + v^2  =  4/49 - 6

u^2 + v^2 =  -290 /  49

1/u^2  + 1/v^2   =  [ u^2 + v^2 ] / (uv)^2  =  [ -290 / 49 ]  / [ 9 ]   =  -290 / 441

3. Not too hard

Area of  AMD + BNC =  (1/2) * 1  =  1/2

Area of MBND  =  Area of square  - 1/2 =    1/2

Shaded area = (1/2)[MBND ]   =  1/2 * 1/2 =  1/4

2.

PQ = sqrt (2^2 + 2^2 )  = sqrt 8

PA = QA =  sqrt [ 7^2 + 5^2] =  sqrt [ 74]

Law of Cosines

PQ^2  = PA^2 QA^2  - 2(PA * QA) cos PAQ

8 = 74 + 74  - 2(74)cosPAQ

[ 8 - 2 * 74 ] / [ -2 * 74 ]  = cos PAQ  =  35 /   37

sin PAQ =  sqrt [ 37^2  - 35^2 ] /  37  =   sqrt [144] / 37  =  12 /  37

1.

                  P

   9                           17

                                 Y

Q                X                   R

             10

Since PX  is a bisector     let  QX = 10 - m  and RX  = m

So

RX / PR  = QX / PQ

m / 17  =  (10 - m)  / 9

9m = 17(10 - m)

9m  = 170 - 17m

26m  = 170

m  = 170 / 26  =  85/13  =XR

Law of Cosines

PQ^2  = QR^2 + PR^2   - 2(QR * PR)cos (PRQ)

9^2  = 10^2  + 17^2  - 2(10 * 17) cos (PRQ)

[ 9^2 - 10^2  - 17^2  ] / [ 340 ] =  cos (PRQ)  = 77/85

sin (PRQ) =  sqrt [ 85^2 - 77^2 ] / 85 =  36/85

Law of Sines

XY / sin (PRQ)   = XR  / sin 90

XY  =  sin (PRQ) (XR)

XY  = (36/85) ( 85/13)  =  36 / 13

Here : https://web2.0calc.com/questions/semicircles_5

Impossible

QR + PR  < PQ

Which violates the Triangle Inequality Theorem

https://web2.0calc.com/questions/angle-problem_3