CPhill

See my answer, here :

https://web2.0calc.com/questions/help_93576

OK...didn't realize that  .....we don't want to encourage  cheating....

OK.....

Prime factor the integer  under  the radical

Call  the  index of the radical, n

If  n  does not evenly divide  every  exponent in the prime factorization, then  we have an irrational result

Two examples

²√ 16  =   ²√2^4      2 divides 4 evenly,,,so we have a rational answer  because   √16  = 4

³√ 32  =  ³√ 2^5      since 3 does not evenly divide 5, we have an irrational answer  =  2³√2^2  = 2³√4

Multiplying  the original equation through by  12  we get

3x  + y =  12      rearrange as

y  =  -3x  + 12

m = -3     b  =12

x intercept....set y=  0

0 = -3x + 12

-12 = -3x

-12/-3 = 4   = x intercept  = a

a + b + m  = 

4  + 12   -  3   =

13

From the center of the  circle (call it O )  connect    OB

Then  OB  = OA  = 18

So angle BAO  =angle ABO   = 42°

Then angle AOB =  180  - 2(42)  = 180 - 84 =  96°  

Using the Law of Cosines

AB^2 = (OA)^2  + (OB)^2   -2 (OA * OB) cos (AOB)

AB^2  =  (18)^2  + (18)^2  - 2(18*18) cos (96)

AB  = sqrt [  (18)^2  + (18)^2  - 2(18*18) cos (96) ]

AB = sqrt (715.7)  ≈   26.75

3^2 + 4^2   = x^2

9  +  16  = x^2

25   =  x^2       take the positive root

5  = x

Note  that rhe line rises from right to left  and  y changes  by  2  when x changes by -1

So.... (9,0)  will  be a lattice point   and (0,18) will be a lattice point

So....the number of lattice points =    [ 9 - 0 ] + 1  =   10 lattice points

Here's a graph : https://www.desmos.com/calculator/drzzupjy3g

Rearange  as

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

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

(x - 1/2)^2  +  ( y -1/2)^2   =   1/2

x will be maximized  when  y = 1/2

So

( x - 1/2)^2   +  (1/2 - 1/2)^2  =  1/2

(x  -1/2)^2    +   0     =   1/2

(x - 1/2)^2  = 1/2           take the positive root

x - 1/2 =  sqrt (1/2)

x - 1/2  =  sqrt (2) / 2

x =  [ 1 + sqrt (2)  ]  / 2

See the  following image  :

B is  point A  reflected over  one of the lines  and C  is  the reflection of A over the other line

Consider FCAB

Angle  CFB = Angle FBA  = Angle GHO = Angle AEG  = Angle CEG =  90°

Angle  OGH = 90 - 38 =  52°  = Angle EGC

So angle ECG  = 90  - 52  =  38°= angle ACF

Then the sum of  the interior angles of FCAB  =  360

So

angle BAC  = 360  -  angle FBA - angle CFB  - angle  ACF

angle  BAC  = 360  - 90  - 90 - 38 =  142°   [ supplemental to the original 38° angle  ]