✨ Simplify Result

lib/screens/calculator_home_page.dart

@@ -300,6 +300,15 @@ class _CalculatorHomePageState extends State<CalculatorHomePage> {
                     ),
                   ),
                 ),
+                Expanded(
+                  child: Tooltip(
+                    message: '未知数二号',
+                    child: FilledButton.tonal(
+                      onPressed: () => _insertSymbol('y'),
+                      child: Text('y', style: GoogleFonts.robotoMono()),
+                    ),
+                  ),
+                ),
               ],
             ),
             const SizedBox(height: 8),

lib/solver.dart

@@ -120,7 +120,7 @@ class SolverService {
         stepNumber: 3,
         title: '求解 x',
         explanation: '两边同时除以 x 的系数 ($newA)。',
-        formula: '\$\$x = \frac{$newD}{$newA}\$\$',
+        formula: '\$\$x = \\frac{$newD}{$newA}\$\$',
       ),
     );
 
@@ -174,6 +174,14 @@ class SolverService {
             formula: factors.solution,
           ),
         );
+        steps.add(
+          CalculationStep(
+            stepNumber: 4,
+            title: '化简结果',
+            explanation: '将分数化简到最简形式，并将负号写在分数外面。',
+            formula: factors.solution,
+          ),
+        );
         return CalculationResult(steps: steps, finalAnswer: factors.solution);
       }
     }
@@ -529,13 +537,29 @@ ${b1}y &= ${c1 - a1 * x}
   ({String formula, String solution})? _tryFactorization(int a, int b, int c) {
     if (a == 0) return null;
     int ac = a * c;
-    for (int i = 1; i <= sqrt(ac.abs()); i++) {
+    int absAc = ac.abs();
-      if (ac % i == 0) {
+    for (int d = 1; d <= sqrt(absAc).toInt(); d++) {
-        int j = ac ~/ i;
+      if (absAc % d == 0) {
-        if (check(i, j, b)) return formatFactor(i, j, a);
+        int d1 = d;
-        if (check(-i, -j, b)) return formatFactor(-i, -j, a);
+        int d2 = absAc ~/ d;
-        if (check(i, -j, b)) return formatFactor(i, -j, a);
+        List<int> signs1 = ac >= 0 ? [1, -1] : [1, -1];
-        if (check(-i, j, b)) return formatFactor(-i, j, a);
+        List<int> signs2 = ac >= 0 ? [1, -1] : [1, -1];
+        for (int s1 in signs1) {
+          for (int s2 in signs2) {
+            int m = s1 * d1;
+            int n = s2 * d2;
+            if (check(m, n, b)) return formatFactor(m, n, a);
+            m = s1 * d1;
+            n = s2 * (-d2);
+            if (check(m, n, b)) return formatFactor(m, n, a);
+            m = s1 * (-d1);
+            n = s2 * d2;
+            if (check(m, n, b)) return formatFactor(m, n, a);
+            m = s1 * (-d1);
+            n = s2 * (-d2);
+            if (check(m, n, b)) return formatFactor(m, n, a);
+          }
+        }
       }
     }
     return null;
@@ -544,32 +568,62 @@ ${b1}y &= ${c1 - a1 * x}
   bool check(int m, int n, int b) => m + n == b;
 
   ({String formula, String solution}) formatFactor(int m, int n, int a) {
-    int common = gcd(n.abs(), a.abs());
+    // Roots are -m/a and -n/a
-    int num = n ~/ common;
+    int g1 = gcd(m.abs(), a.abs());
-    int den = a ~/ common;
+    int root1Num = -m ~/ g1;
+    int root1Den = a ~/ g1;
 
-    final a1 = den;
+    int g2 = gcd(n.abs(), a.abs());
-    final c1 = num;
+    int root2Num = -n ~/ g2;
-    final a2 = a ~/ den;
+    int root2Den = a ~/ g2;
-    final c2 = m ~/ a2;
 
-    final f1Part1 = a1 == 1 ? 'x' : '${a1}x';
+    String sol1 = _formatFraction(root1Num, root1Den);
-    final f1 = c1 == 0 ? f1Part1 : '$f1Part1 ${c1 >= 0 ? '+' : ''} $c1';
+    String sol2 = _formatFraction(root2Num, root2Den);
 
-    final f2Part1 = a2 == 1 ? 'x' : '${a2}x';
+    // For formula, show (a x + m)(x + n/a) or simplified
-    final f2 = c2 == 0 ? f2Part1 : '$f2Part1 ${c2 >= 0 ? '+' : ''} $c2';
+    String f1 = a == 1 ? 'x' : '${a}x';
+    f1 = m == 0 ? f1 : '$f1 ${m >= 0 ? '+' : ''} $m';
 
-    final int x1Num = -c1, x1Den = a1;
+    String f2;
-    final int x2Num = -c2, x2Den = a2;
+    if (n % a == 0) {
+      int coeff = n ~/ a;
+      f2 = 'x ${coeff >= 0 ? '+' : ''} $coeff';
+      if (coeff == 0) f2 = 'x';
+    } else {
+      f2 = 'x ${n >= 0 ? '+' : ''} \\frac{$n}{$a}';
+    }
 
-    final sol1 = x1Den == 1 ? '$x1Num' : '\\frac{$x1Num}{$x1Den}';
+    String formula = '\$\$($f1)($f2) = 0\$\$';
-    final sol2 = x2Den == 1 ? '$x2Num' : '\\frac{$x2Num}{$x2Den}';
 
-    final solution = x1Num * x2Den == x2Num * x1Den
+    String solution;
-        ? 'x_1 = x_2 = $sol1'
+    if (root1Num * root2Den == root2Num * root1Den) {
-        : 'x_1 = $sol1, \\quad x_2 = $sol2';
+      solution = '\$\$x_1 = x_2 = $sol1\$\$';
+    } else {
+      solution = '\$\$x_1 = $sol1, \\quad x_2 = $sol2\$\$';
+    }
 
-    return (formula: '\$\$($f1)($f2) = 0\$\$', solution: '\$\$$solution\$\$');
+    return (formula: formula, solution: solution);
+  }
+
+  String _formatFraction(int num, int den) {
+    if (den == 0) return 'undefined';
+
+    // Handle sign: make numerator positive, put sign outside
+    bool isNegative = (num < 0) != (den < 0);
+    int absNum = num.abs();
+    int absDen = den.abs();
+
+    // Simplify fraction
+    int g = gcd(absNum, absDen);
+    absNum ~/= g;
+    absDen ~/= g;
+
+    if (absDen == 1) {
+      return isNegative ? '-$absNum' : '$absNum';
+    } else {
+      String fraction = '\\frac{$absNum}{$absDen}';
+      return isNegative ? '-$fraction' : fraction;
+    }
   }
 
   int gcd(int a, int b) => b == 0 ? a : gcd(b, a % b);