Finding roots
1.  Bisection method

f[x_] :=  3 (x - 1/7) (x - 3/19) (x - 1.85)

f[1/7]

f[3/19]

f[1.85]

f[x]

Expand[%]

Factor[%]

Plot[f[x], {x, -1, 2}]

Plot[f[x], {x, 0, 0.5}]

Plot[f[x], {x, 0.1, 0.2}]

Bisection Method

a  =  0;
b = 2;
mid  =  
        a  +  (b - a)/
            2;

fa  = f[a]

fb = f[b]

fm = f[mid]

For[i = 1,  i ≤ 10, 
      i = i + 1, 
If[ fa fm > 0,  a  =  mid; 
        fa  =  fm;,  b  =  mid; 
        fb =  fm;]; 
mid  =  
        a  +  (b - a)/2; 
fm  =  
        f[mid];]

fm

N[mid]

Bisection : Version 2

a  =  0;
b = 2;
mid  =  a  +  (b - a)/2;
fa  = f[a];
fb = f[b];
fm = f[mid];
xtol  =  0.001;
ytol  =  0.0001;
\(While[\((b - a\  > \ xtol)\) && \ \((Abs[fm]\  > \ ytol)\), 
      If[\ fa\ fm > 0, \ a\  = \ mid; fa\  = \ fm;, \ b\  = \ mid; 
        fb = \ fm;]; 
mid\  = \ 
        a\  + \ \((b - a)\)/2; 
fm\  = \ 
        f[mid];]
\ 
\)

N[mid]

Abs[fm]

N[b - a]