CPhill

Simplify as

4x^2 + nx + 104 <  0

Set up as an equality

4x^2 + nx + 104 =  0

This will have real solutions when

n^2  - 4 (4) (104) ≥  0

n^2 ≥ 1664

n ≥ 40.79            or   n  ≤ -40.79

So....for our purposes   n < 40.79     and  n > -40.79

The number of  possible values for n  =  [ -40 , 40 ] =   81 possible values

x^2 + y^2   = r^2

(x + 2)^2 + ( y  -1)^2   = r^2            set these equal  for r^2

x^2 + y^2 =  x^2 + 4x +4 + y^2 - 2y + 1

4x - 2y  = -5     (1)

And

x^2 + y^2 = r^2

(x +3)^2 + ( y  -2)^2  = r^2           set these equal for r^2

x^2 + y^2  =  x^2 + 6x + 9 + y^2 - 4y + 4

6x - 4y =  -13      (2)

Multiply (1)  throughby -2  and add to (2)   and we have

-2x = -3

x =3/2

To find y, use (1)

4(3/2)  - 2y = -5

6 - 2y = -5

11 = 2y

y = 11/2

(x, y) = ( 3/2 , 11/ 2)

The vertex is  ( 3,1)  and the points  (0,7) and (6,7) are on the graph

c = 7

1 = a (3)^2  + b(3) + 7

7 = a(6)^2 + b(6) + 7            simplify

9a + 3b =  - 6    →  3a + b = -2 →   b = -2 - 3a     (1)

36a + 6b =  0       (2)

Sub (1)  into (2)

36 a + 6(-2-3a)  = 0

36a - 12 - 18a  =  0

18a =  12

a = 2/3

b = -2  - 3(2/3)  =  -2 - 2 =  -4

a * b * c =    (2/3) * (-4) * 7   =   -56 / 3

The axis of symmetry is 2

If   (4,7)   lies on the graph  then   (2 + 2 , 7)  =  (4, 7)

So

(2 - 2, 7)  =  (0, 7)  also lies  on the  graph

Sorry....see corrections

Simplify as

x^2 -6x + y^2 +10y  =  12 + 6               complete the square on x, y

x^2 - 6x + 9 + y^2 + 10y + 25 =  12  + 6+ 9 + 25

(x - 3)^2  + (y + 5)^2  =  52

Ths  is a  circle with  r^2  = 52

Area =   pi * r^2   =  52 pi

Law of Cosines

18^2  =  12.5^2 + PM^2 - 2 ( 12.5 * PM)  cos (QMP)

23^2  = 12.5^2 + PM^2 - 2(12.5 * PM)  (-cos (QMP)

Simplify

(18^2 - 12.5^2 - PM^2 ) /  [ -25 PM ]   = cos (QMP)

[23^2 - 12.5^2 - PM^2 ] / [ 25 PM =  cos (OMP)

Equate cosines  and simplify

- [ 167.75 - PM^2 ]   =  [372.75 - PM^2]

2PM^2  = 540.5

PM^2  = 270.25

PM =sqrt [ 270.25]  ≈  16.44

https://web2.0calc.com/questions/triangles_143

****

r = 3sin θ

r = 3 (y / r)

r^2 = 3y

x^2 + y^2  = 3y

x^2 + y^2 - 3y  = 0            complete the square on  y

x^2 + y^2 - 3y + 9/4  = 9/4

x^2 + ( y - 3/2)^2  = 9/4

https://web2.0calc.com/questions/angle-bisectors_37

https://web2.0calc.com/questions/angle-bisector_13

Inradius =  area / semi-perimeter  =

[ 1/2 * a^2 * sqrt (3) /2 ]  / [ 3a/2]  =

[ sqrt (3) / 4 * a^2 ] / [  3a/ 2 ]  =

a / (2 sqrt 3)

sqrt (3) a  / 6

https://web2.0calc.com/questions/triangle_62177

                             A

                     7        12           17

             B                    M                      C

Law of Cosines

12^2 =  7^2  + (BC/2)^2  - 2 (7 * BC /2) cos ( ABC)

17^2  = 7^2 + BC^2 - 2 (7 * BC) cos (ABC)

Equate cosines and simplify

[144 - 49 - BC^2/4] / [ 7 BC ]   =   [289 - 49 - BC^2 ] / [  14 BC]

[ 95  - BC^2 / 4 ]  / 7  =  [ 240 - BC^2 ] / 14

[95 - BC^2/4 ]  = [ 240 - BC^2 ] / 2

95 -BC^2 / 4 = 120 - BC^2/2

120 - 95  = BC^2 / 4

25 = BC^2  /4

BC^2 = 25 * 4

BC = 5 * 2  = 10

Impossible  because  of the trianle inequality,  AB + BC   > AC

But

AB + BC  = 

7 + 10  not greater than  17

https://web2.0calc.com/questions/counting_81244

https://web2.0calc.com/questions/help-me_12712

(a + 2)  / ( a + 1)  =  b / 9

8 (a + 2)  = b ( a + 1)

8a + 16  = ab + b

ab + b   - 8a   = 16               multiply the coefficients on a,b and  add  to  both sides

ab + b - 8a - 8  = 16 - 8

(a + 1) ( b - 8)  =  8

Factors of 8                a           b

4, 2                             3         10

2, 4                             1         12

1, 8                             0         16

8. 1                             7          9

-4 , -2                         -5         6

-2, - 4                         -3         4

-1, -8                          -2         0

-8. -1                          -9         7