CPhill

Shortest altitude is drawn to  the longest side

Area (using Heron's Formula)  = sqrt [ 17 * 2 * 7 * 8 ] =   sqrt [ 1904]

[ABC ]   = (1/2) AB * shortest altitude

sqrt [ 1904 ] = (1/2) * 15 * shortest altitude

sqrt [ 1904 ] / 7.5  =  shortest altitude ≈   5.82

x^2 + y^2 = 25

(4)^2  + y^2  = 25

y^2  = 9

y = -3       y = 3

A= (4,-3)     B = (4,3)

AB =  (3 - -3)  =  6

https://web2.0calc.com/questions/right-triangles_33

a + b =1         square both sides

a^2 + 2ab + b^2  = 1

a^2 + b^2 + 2ab = 1

2 + 2ab = 1

2ab = -1

ab = -1/2

-ab =1/2

a^3 + b^3 =  (a +b) ( a^2 + b^2 - ab)

a^3 + b^3  = ( 1) ( 2 + 1/2)  =  5/2

a11 - a10 = a9

1 - 4  = a9

-3 = a9 

a10 - a9 =a8

4 - -3 = a8

7 = a8

a9  - a8  = a7

-3 - 7  = a7

-10 = a7

a8  - a7  = a6

7 -  -10  = a6

17 = a6

Simplify as

3x^2 - kx + 33 = 0

This will have no real solutions when the discriminant is < 0

So

k^2  - 4(3) (33) < 0

k^2  - 396 < 0

k^2 < 396

k < sqrt (396)

k = 19

a + ab =   250

a -  ab =  -240          add these

2a = 10

a = 5

5 + 5b = 250

5b = 245

b = 245 / 5  =   49

(a, b)  = (5 , 49)

191 is prime

phi (191)  = 191 - 1  =  190 

1200 =  12 * 5^2 * 2^2  = 2^4 * 3 * 5^2

Using Euler's Totient Function

phi (1200)  =    1200 / ( 2 * 3 * 5) *  (2-1)(3-1) (5-1) = 40 (1)(2)(4)  =  320

Thx, NTS....here's another way  { no square roots involved !!! }

x^2 + 6x - 2  = 0

In the form mx^2 + nx + p  = 0

Product of the roots ab = -n/m =   -2  

So 2ab = -4

Sum of the roots a + b =  p/m =  -6     square both sides

a^2 + b^2 + 2ab = 36

a^2 + b^2 - 4 =36

a^2 + b^2  = 40

1/a^3  + 1/b^3  =

(a^3 + b^3) / (ab)^3           Note  :  a^3 + b^3 = (a + b) (a^2 + b^2 - ab)

So we have

(a + b) ( a^2 + b^2 -ab)  / (ab)^3  =

(-6) (40 - -2) / (-8)   = 

(6) (42 / 8)  =   31.5