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Well, we knw that AE is twice as big as ED, and AE + ED = AD. 

We can wwrite an equation:

2x + x = 30, where 2x is AE and x is ED.

3x = 30

x = 10

AE = 2x, so:

AE = 20

Answer: 20

So basically the ratio between their speeds is 19 to 18, edith 19, so when edith finishes 19/19 of the race, foley finishes 18/19, meaning there is 1/19 left to go. 1/19 x 24 = 24/19

PLug in the quadratic formula:\(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\)

\(1 + \frac{3}{6} + \frac{5}{6^2} + \frac{7}{6^3} \)

9/6  +   5/36  + 7/216

[ 324 + 30 + 7 ] / 216

[ 361 ] / [ 216]

When Edith runs  (24 - 5) miles   =19 miles, Foley has run (24 - 6)  = 18 miles

So.....Foley  runs at   (18/19)  of Edith pace

When she finishes Foley has run    (18/19) (24) =  432/19  km

So....he is    24  - 432 / 19  =  24/19 km  behind

\(\frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 10} \)

1/4  + 1/28 + 1/70                   

4 = 2^2

28   = 2^2 * 7

70  =  2 * 5 * 7                common  denominator  =  2^2 * 5 * 7  =  140

   35/ 140  +  5  /140  +  2 /140  =  

[35 + 5 + 2 ]  / 140  =

42 / 140  = 

3 / 10

 We have triangle  MNZ

Angle ZNM  + Angle ZMN + Angle NZM  =  180°

And we have triangle  PNM

Angle PNM + Angle PMN  + NPM  =   180°

So....the sum  of all the given angles =  360°

Not  sure about this  but notice      (NP =not possible)

 b^2  - a^2     b^2   a^2

   1                 1      0       (violates  our rules.... a,b must be positive)

   2                    NP

   3                 2      1

   4                 2      0      (violates our  rules)               

   5                 3      2

   6                   NP

   7                4       3

   8                3       1

   9                5       4

  10                 NP

  11               6       5

  12               4       2

  13               7       6

  14                NP

  15                8      7    

  16                5      3

  17                9      8

  18                NP

It appears that all odds (except 1) can  be made  

And it appears that  all  evens can  be made except  4 and  the ones of the form 2 + 4(n -1)   where n =  1,2,3,4....

You should check  for yourself  with a few values !!!

Note that  when n = 506     2 + 4 (506 - 1) =  2022

So.....the answer  should  be     2 values = (1,4) + 506 values =   508 integers cannot be made

Hope that helps....

                                        C

                            E                    D

                                      H 144

              A 66                                         B

Consider triangle    AHB     formed by BE and AD

Angle BAH = 66

Angle AHB  =144

But the sum of these > 180   which is  impossible

No solution

A + B + C  = 180

So

(p + 2q) + (6p -5q) + (14p + 8q)  =  180

21p + 5q  =  180

21p  = 180  - 5q

p =   [ 180  - 5q ]  /  21

Simplify as

x^2 - mx  + 14  = 0

Possible factorizations

(x - 2) (x -7)  = 0             m  =  -9

(x + 2) (x + 7)  = 0          m  =  9

(x  -14) (x -1)  =  0          m    =  -15

(x + 14) ( x + 1)  =  0      m  =  15

In the first function , x  cannot =  0

In the second function

5x + 6 -11 ≥  0

5x - 5 ≥  0

5x ≥ 5

x ≥ 1

So.......the most restrictive interval is the  second one

So the domain  is  [1, inf )

A

12sqrt 3          X

B                12                  C

Hypotenuse  = 12 * 2   = 24

Let AX =  x

Let CX =  24 -x 

Since BX is an angle  bisector

AX / AB =  CX / AB

x / (12sqrt 3)  = (24 - x)  /12

x /sqrt 3  =  24 - x

x =  (24 - x) sqrt 3

x = 24sqrt 3  -  sqrt (3)x

x + sqrt 3  = 24 sqrt 3

x ( 1 + sqrt 3) =  24 sqrt (3)

x =  [ 24sqrt (3) ] / [ 1 + sqrt 3]

x = [24 sqrt 3 ] [1 - sqrt 3] / [ 1 -3]

x = -12 [ 1 - sqrt 3]   =  12 [ sqrt (3) - 1]  = AX

[BXA  ]    = (1/2) (AB)(AX) sin (30°)  =   (1/2)(12sqrt (3))(12[sqrt (3) - 1]) (1/2)  =

36 [sqrt (3) ] [ sqrt (3)   - 1]  = 

36 ( 3 - sqrt 3 )   ≈ 45.65

Simplify as

x^2 + 12x - 4 =  0

Call the roots a,b

Product of  roots

ab  = -4

2ab  =  -8

Sum of  roots 

a + b  =  -12      square both sides

a^2 + 2ab + b^2 = 144

a^2 + b^2  -8   =144

a^2 + b^2   = 152

Re-arrange to 

x^2 +12x -4 = 0     complete the square 

(x+6)^2  = 4 + 36 

x+6 = +- sqrt 40 

x =  - 6 +- sqrt 40      <==== these are the roots 

Now square the roots

(-6 + sqrt 40)² = 76 - 12 sqrt 40 

(-6  - sqrt 40)²  = 76 +12 sqrt 40

                        added together =  76 + 76 =   + 152  

Well we know that x can not equal 0, because then we would have a number that is undefined, so the domain of the function f(x) is 

( - inf , 0 ) U ( 0 , inf )

Answer: ( - inf , 0 ) U ( 0 , inf )

\(2x^2+4x-1=x^2-8x+3\)

We can make this is the form  

\(ax^2+bx+c=0\)

We get:

\(x^2+12x-4=0\)

a = 1, b = 12, and c = -4.

Using the quadratic formula \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\), we get:

\(x = {-12 \pm \sqrt{12^2+16} \over 2}\)

\(x = {-12 \pm \sqrt{160} \over 2}\)

\(x = {-12 - \sqrt{160} \over 2}\), \(x = {-12 + \sqrt{160} \over 2}\)

Now we can square both values of x and then add them:

-151.894663844 + 0.10533615595 = -151.789327688

Answer: -151.789327688

Start with a list of primes > 20 

  23    29  31   37   

23 * 37 is too large 

so you have 

  23 x 29    = 667

   23 x 31   = 713 

          and that's it !     

A = 0.67

B = 1536

C = 0.32

D = -16

E = 4.24

Smallest to biggest:

D, C, A, E, B, or -16, 0.32, 0.67, 4.24, 1536

Answer: D, C, A, E, B

Answer: -16, 0.32, 0.67, 4.24, 1536

Well, we can see that this is a special right triangle, a 30-60-90 triangle, where m angle A = 30, m angle C = 60, and m angle B = 90.

The side lengths are in the ratio 1:sqrt3:2, which in our case is 12:12 x sqrt3:24

12 x sqrt 3 = 20.7846096908 or approx 20.8 as shown in the graph below:

ALERT:     Sorry, the top point is meant to be A, the bottom left one B, and the bottom right one C.

We can multiply 12x 20.7846 x 1/2 and get:

124.7076

or approximately

124.71

Answer: 124.71

Triangles CDE  and EFB  are similar

So

CD / DE  =  EF / FB

(1 - x)  / x   =  x / (3 - x)             cross-multiply

(3 - x) ( 1 -x)   = x^2

x^2  - 4x + 3   =   x^2

-4x + 3  = 0

3  = 4x

x  = 3/4

Area of  square  = x^2  =   (3/4)^2  =  9 / 16

Notice something kinda' neat   

Side of square  =  x =  (product of   legs)  / (sum of legs) =  (3 * 1) / ( 3 + 1)  = 3 / 4

       A                          B

                                   4

                                   P

                                   1

     D     1      Q     4    C

P  = (5 , 1)

Q =  (1,0)

A  = (0, 5)

PA  =   sqrt 41

PQ  =  sqrt  17

AQ = sqrt 26

Law of Cosines

PQ^2  = AQ^2  + PA^2  - 2(AQ * PA) * cos (PAQ)

17  =  26 + 41 -  2 ( sqrt 26 * sqrt 41) cos (PAQ)

cos (PAQ) =   [ 17 - 26 - 41 ] / [ - 2 sqrt 1066 ]  =    -50 / [ -2 sqrt 1066 ]  =   25 / sqrt (1066)

sin (PAQ)  = sqrt [  1 - (25 / sqrt 1066)^2 ] =  sqrt  [ 1 - 625 / 1066 ]  = sqrt [ 441 / 1066 ] =

21 / sqrt 1066  =   (21 sqrt (1066) / 1066

x int  =  (x , 0)

8 = [ 14 - 0 ] / [ 0 - x ]

8  =  14 / -x

-x  = 14/8

x = -14/8  =  -7/4 =  x intercept

ab + 4b   =  131 - 3a

b ( a + 4)  = ( 131 - 3a)

b  = (131 - 3a)  / ( a + 4)

a       b

7      10

9       8

Min value  =  l 9  - 8 l  = 1