diff --git a/README.md b/README.md
index c00f452..11abbcb 100644
--- a/README.md
+++ b/README.md
@@ -63,19 +63,176 @@ this project uses submodules (maybe recursively), so either :
       -> Δ == 0 -> 1 solution  : x = ( -b / 2a )
       -> Δ < 0  -> 2 solutions : x = ( -b / 2a ) +- i( √|Δ| / 2a )
       -> solution : 
-        - a;                       // double
-        - b;                       // double
-        - c;                       // double
         - delta_sign;              // + or -
         - delta_absolute;          // |Δ|
+        - delta_sqrt;              // √|Δ|
+
         - first_term_gcd;          // gcd(b, 2a)
         - first_term_numerator;    // -b / gcd
         - first_term_denominator;  // 2a / gcd
         - first_term;              // double (-b / 2a)
+
         - second_term_gcd;         // gcd(√|Δ|, 2a)
         - second_term_numerator;   // √|Δ| / gcd
         - second_term_denominator; // 2a / gcd
         - second_term;             // double (√|Δ| / 2a)
+
         - double solution1;        // first_term + second_term
         - double solution2;        // first_term - second_term
 6. print solution
+
+---
+
+# sqrt implementation
+
+finding the square root of x
+
+## dichotomy method
+
+```
+     solution
+        ↓
+|------------------------------ 1. define range bound_1 : 0
+|-----------------------------| 2. choose range bound_2 : x
+|--------------+--------------| 3. take mid : (bound_1 + bound_2) / 2
+|--------------|-------ø------| 4. choose range bound : bound_1 or bound_2
+
+---------------|--------------- 1. range bound_1 = mid
+|--------------|--------------- 2. range bound_2 = chosen previous bound
+|------+-------|--------------- 3. mid
+|--ø---|-------|--------------- 4. choose range
+
+-------|----------------------- 1. range bound_1 = mid
+-------|-------|--------------- 2. range bound_2 = chosen previous bound
+-------|---+---|--------------- 3. mid
+-------|---|-ø-|--------------- 4. choose range
+
+-----------|------------------- 1. range bound_1 = mid
+-------|---|------------------- 2. range bound_2 = chosen previous bound
+-------|-+-|------------------- 3. mid
+-------|-|ø|------------------- 4. choose range
+
+---------|--------------------- 1. range bound_1 = mid
+-------|-|--------------------- 2. range bound_2 = chosen previous bound
+-------|+|--------------------- 3. mid
+-------ø|---------------------- 4. choose range
+
+--------|---------------------- --> solution
+```
+
+
+## Newton–Raphson method
+
+it's like a self-correcting binary search, we get rid of the step "choose range" : 
+
+```
+     solution
+        ↓
+------------------------------| 1. define range bound_1 : v
+--|---------------------------| 2. choose range bound_2 : x/v
+--|-------------+-------------| 3. take mid : (v + x/v) / 2
+
+----------------|-------------- 1. range bound_1 : v = mid
+------|---------|-------------- 2. range bound_2 : x/v
+------|----+----|-------------- 3. mid
+
+-----------|------------------- 1. range bound_1 : v = mid
+-|---------|------------------- 2. range bound_2 : x/v
+-|----+----|------------------- 3. mid
+
+------|------------------------ 1. range bound_1 : v = mid
+------|-------|---------------- 2. range bound_2 : x/v
+------|---+---|---------------- 3. mid
+
+----------|-------------------- 1. range bound_1 : v = mid
+------|---|-------------------- 2. range bound_2 : x/v
+------|-+-|-------------------- 3. mid
+
+--------|---------------------- --> solution
+```
+
+### mathematical proof that each range is automatically in the right range :
+  - if the value was higher than the answer, then new value is below old value, and vice versa
+  - how ? : 
+    - define `x`, solution `s = √x`, and value `v = (old_value + x / old_value) / 2`
+    - supposition : if `v < s` , then `new_v > v`, else `new_v < v` : 
+    - demonstration : 
+      1. if `v < s` : 
+            v < s
+        <=> v < √x
+        <=> v² < x (that's actually how we know that v < s)
+        <=> v²/v < x/v
+        <=> v < x/v
+        -> and is s < x/v ? : 
+            v < s
+        <=> v < √x
+        <=> v² < x (as previously)
+        <=> v² < x²/x
+        <=> v² * x < x²
+        <=> (v * √x)² < x²
+        <=> v * √x < x
+        <=> v * √x < v * x/v
+        <=> √x < x/v
+        -> so indeed : if v < √x, then v < √x < x/v == v < s < x/v
+        -> conclusion, the new range < v , x/v > contains the solution
+      2. the same demonstration works for `v > s`
+
+### here is a more intuitive demonstration, with x = 20 : 
+
+1. **show that if `v² > x` (== `v > s`) then `v > s > x/v`, and if `v² < x` (== `v < s`) then `v < s < x/v` :**
+  1.1. **for value too high `v > s` :**
+    1.1.1 **why `v > x/v` :**
+      - let's take initial value v = 5 :
+      - is 5² the solution ? 5² == 25 -> so no, 5 is not the sqrt, it's too high
+        ```
+                  v                   v²
+             0   (5)   10   15   20   25
+        v  : |----|----|----|----|----|
+        x/v: |---|---|---|---|---| <----- squiz it, so the previous 5 portions fit the x = 20 size
+             0  (4)  8   12  16  20
+                x/v              x
+        ```
+      - the value of the new portion is 4, and we can visually see that it's lower than the previous portion 5
+      - so : `v > x/v`
+      1.1.2 **why `s > x/v` :**
+      - let's take the value v = 5 : 
+      - we already showed that it's too high, now we will see that x/v == 20/5 is too low : 
+        ```
+                     v
+                0   (5)   10   15   20   25
+        v  :    |----|----|----|----|----|
+        x/v:    |---|---|---|---|---|   <----- squizz
+                0  *1  *2  *3  *4  *5       -> number of portions
+                01234                       -> portion size
+        (x/v)²: |---|---|---|---|
+                0   4   8   12  16
+        ```
+      - the portion size is smaller than the number of portions, so it's too small to be the sqrt, indeed we visually see that this portion size `x/v` is a root of a smaller number : `(x/v)² == 16`
+      - so : `s > x/v`
+    1.1.3. **conclusion :**
+      -     v > s
+      - and v > x/v (<- this proof is not essential)
+      - and s > x/v (<- we actually only need this proof)
+      - so `v > s > x/v`
+  
+  1.2. **for value too high `v < s` :**
+    - this is the same demonstration but in other direction, let's just summarize it :
+    - let's take initial value v = 4 :
+      ```
+                      v           v²
+                  0  (4)  8   12  16
+      (1) v  :    |---|---|---|---|
+      (2) x/v:    |----|----|----|----|   -----> stretch
+      (3)         0   (5)   10   15   20
+                      x/v             x
+      (4)         0   *1   *2   *3   *4       -> number of portions
+                  012345                      -> portion size
+      (5) (x/v)²: |----|----|----|----|----|
+                  0    5    10   15   20   25
+      ```
+    - (1) : 4 is not the sqrt of x == 20, it's too smalle : (4² == 16) < 20
+    - (2) : stretch it, so the previous 4 portions fit the x = 20 size
+    - (3) : the new portion x/v == 5, is more than v == 4, so `v < x/v`
+    - (4) : portion size is bigger than number of portions, so it's too big to be the root
+    - (5) : indeed, we see that the portion² == (x/v)² is bigger than x, so √x < x/v == `s < x/v`
+    - so `v < s < x/v`
diff --git a/headers/computorv1.h b/headers/computorv1.h
index baebff8..2cc9e44 100644
--- a/headers/computorv1.h
+++ b/headers/computorv1.h
@@ -132,14 +132,18 @@ typedef struct
 {
     e_delta_sign delta_sign;     // DELTA_PLUS or DELTA_MINUS or DELTA_ZERO
     double delta_absolute;       // |Δ| == |b² - 4ac|
+    double delta_sqrt;           // √|Δ|
+                                 //
     int first_term_gcd;          // gcd(b, 2a)
     int first_term_numerator;    // -b / gcd
     int first_term_denominator;  // 2a / gcd
     double first_term;           // double (-b / 2a)
+                                 //
     int second_term_gcd;         // gcd(√|Δ|, 2a)
     int second_term_numerator;   // √|Δ| / gcd
     int second_term_denominator; // 2a / gcd
     double second_term;          // double (√|Δ| / 2a)
+                                 //
     double solution1;            // first_term + second_term
     double solution2;            // first_term - second_term (not if DELTA_ZERO)
 } s_solution_degree_2;
diff --git a/libft b/libft
index 8edf428..ae9787b 160000
--- a/libft
+++ b/libft
@@ -1 +1 @@
-Subproject commit 8edf428fed1e5e292b4539a0b3e1bfab1c3bd392
+Subproject commit ae9787ba6db99412ff4c6d0bde357e060f350096
diff --git a/src/solver.c b/src/solver.c
index f155b88..8c2f1a2 100644
--- a/src/solver.c
+++ b/src/solver.c
@@ -71,8 +71,6 @@ static int find_gcd(double numerator, double denominator)
 /*
 typedef struct
 {
-    double a;                 // a in "ax + b"
-    double b;                 // b in "ax + b"
     int solution_gcd;         // gcd(b, a)
     int solution_numerator;   // -b / gcd
     int solution_denominator; //  a / gcd
@@ -82,19 +80,20 @@ typedef struct
 /*
 typedef struct
 {
-    double a;                    // a in "ax² + bx + c"
-    double b;                    // b in "ax² + bx + c"
-    double c;                    // c in "ax² + bx + c"
     e_delta_sign delta_sign;     // DELTA_PLUS or DELTA_MINUS or DELTA_ZERO
     double delta_absolute;       // |Δ| == |b² - 4ac|
+    double delta_sqrt;           // √|Δ|
+
     int first_term_gcd;          // gcd(b, 2a)
     int first_term_numerator;    // -b / gcd
     int first_term_denominator;  // 2a / gcd
     double first_term;           // double (-b / 2a)
+
     int second_term_gcd;         // gcd(√|Δ|, 2a)
     int second_term_numerator;   // √|Δ| / gcd
     int second_term_denominator; // 2a / gcd
     double second_term;          // double (√|Δ| / 2a)
+
     double solution1;            // first_term + second_term
     double solution2;            // first_term - second_term (not if DELTA_ZERO)
 } s_solution_degree_2;
@@ -107,11 +106,16 @@ static void solve_degree_2(s_solution_degree_2 *solution, double a, double b, do
     delta = b * b - 4 * a * c;
     solution->delta_sign = ft_fsign(delta);
     solution->delta_absolute = ft_fabs(delta);
+    solution->delta_sqrt = ft_sqrt(solution->delta_absolute);
+
+    // first term
     solution->first_term_gcd = find_gcd(b, 2 * a);
     solution->first_term_numerator = -b / solution->first_term_gcd;
     solution->first_term_denominator = 2 * a / solution->first_term_gcd;
     solution->first_term = -b / (2 * a);
-    // solution->second_term_gcd = ;
+
+    // second term
+    // solution->second_term_gcd = find_gcd(b, 2 * a);
     // solution->second_term_numerator = ;
     // solution->second_term_denominator = ;
     // solution->second_term = ;
diff --git a/src/utils/errors.c b/src/utils/errors.c
index 6db650d..1014b62 100644
--- a/src/utils/errors.c
+++ b/src/utils/errors.c
@@ -76,15 +76,16 @@ static void print_context_solution()
     if (solution_g_err->degree == 2)
     {
         solution_2 = solution_g_err->solution_degree_2;
-        dprintf(STDERR_FILENO, "degree 2               : delta > 0                 ( -b/2a +-  √Δ/2a )\n");
-        dprintf(STDERR_FILENO, "                         delta == 0                ( -b/2a )\n");
+        dprintf(STDERR_FILENO, "degree 2               : delta > 0                 ( -b/2a +-  √Δ/2a   )\n");
+        dprintf(STDERR_FILENO, "                         delta == 0                ( -b/2a             )\n");
         dprintf(STDERR_FILENO, "                         delta < 0                 ( -b/2a +- i√|Δ|/2a )\n");
         dprintf(STDERR_FILENO, "delta_sign             : %25s\n", delta_sign_to_str(solution_2.delta_sign));
-        dprintf(STDERR_FILENO, "delta_absolute         : %25g (|b² - 4ac|)\n", solution_2.delta_absolute);
+        dprintf(STDERR_FILENO, "delta_absolute         : %25g ( |Δ| == |b² - 4ac| )\n", solution_2.delta_absolute);
+        dprintf(STDERR_FILENO, "delta_sqrt             : %25g (√|Δ|               )\n", solution_2.delta_sqrt);
         dprintf(STDERR_FILENO, "first_term_gcd         : %25i\n", solution_2.first_term_gcd);
-        dprintf(STDERR_FILENO, "first_term_numerator   : %25i (-b / gcd)\n", solution_2.first_term_numerator);
-        dprintf(STDERR_FILENO, "first_term_denominator : %25i (2a / gcd)\n", solution_2.first_term_denominator);
-        dprintf(STDERR_FILENO, "first_term             : %25g (-b / 2a)\n", solution_2.first_term);
+        dprintf(STDERR_FILENO, "first_term_numerator   : %25i (-b / gcd           )\n", solution_2.first_term_numerator);
+        dprintf(STDERR_FILENO, "first_term_denominator : %25i (2a / gcd           )\n", solution_2.first_term_denominator);
+        dprintf(STDERR_FILENO, "first_term             : %25g (-b / 2a            )\n", solution_2.first_term);
         // dprintf(STDERR_FILENO, "second_term_gcd        : %25g\n", solution_2.second_term_gcd);
         // dprintf(STDERR_FILENO, "second_term_numerator  : %25g\n", solution_2.second_term_numerator);
         // dprintf(STDERR_FILENO, "second_term_denominator: %25g\n", solution_2.second_term_denominator);