diff --git a/lib/solver.dart b/lib/solver.dart index a7defdd..9ba1ee6 100644 --- a/lib/solver.dart +++ b/lib/solver.dart @@ -24,6 +24,65 @@ class SolverService { return formatted; } + /// 获取整数的所有因数（包括正负） + List _getAllDivisors(int n) { + List divisors = []; + int absN = n.abs(); + for (int i = 1; i <= absN; i++) { + if (absN % i == 0) { + divisors.add(i); + if (i != absN) divisors.add(-i); + } + } + return divisors; + } + + /// 尝试对二次方程进行因式分解 + String? _tryFactorQuadratic(double a, double b, double c) { + if (a != a.round() || b != b.round() || c != c.round()) return null; + int aa = a.round(), bb = b.round(), cc = c.round(); + if (aa == 0) return null; // 不是二次方程 + var divisorsA = _getAllDivisors(aa); + var divisorsC = _getAllDivisors(cc); + for (int p in divisorsA) { + int r = aa ~/ p; + for (int q in divisorsC) { + int s = cc ~/ q; + if (p * s + q * r == bb) { + String factor1 = _formatFactorTerm(p, q); + String factor2 = _formatFactorTerm(r, s); + return '($factor1)($factor2)'; + } + } + } + return null; + } + + /// 格式化因式中的项 + String _formatFactorTerm(int coeff, int constTerm) { + String result = ''; + if (coeff != 0) { + if (coeff == 1) + result += 'x'; + else if (coeff == -1) + result += '-x'; + else + result += '${coeff}x'; + } + if (constTerm != 0) { + if (result.isNotEmpty) { + if (constTerm > 0) + result += ' + $constTerm'; + else + result += ' - ${-constTerm}'; + } else { + result += constTerm.toString(); + } + } + if (result.isEmpty) result = '0'; + return result; + } + /// 主入口方法，识别并分发任务 CalculationResult solve(String input) { // 预处理输入字符串 @@ -330,14 +389,67 @@ class SolverService { ), ); - steps.add( - CalculationStep( - stepNumber: 2, - title: '选择解法', - explanation: '我们选择使用配方法。', - formula: r'配方法：$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$', - ), - ); + final factored = _tryFactorQuadratic(a, b, c); + if (factored != null) { + steps.add( + CalculationStep( + stepNumber: 2, + title: '选择解法', + explanation: '我们选择使用因式分解法。', + formula: '\$\$ax^2 + bx + c = 0\$\$', + ), + ); + steps.add( + CalculationStep( + stepNumber: 3, + title: '因式分解', + explanation: '将二次方程分解为两个一次因式的乘积。', + formula: '\$\$$factored = 0\$\$', + ), + ); + final discriminant = b * b - 4 * a * c; + final roots = _calculateSymbolicRoots( + a, + b, + discriminant, + _getSymbolicSquareRoot(discriminant), + ); + steps.add( + CalculationStep( + stepNumber: 4, + title: '解出 x', + explanation: '分别令每个因式为零，解出 x 的值。', + formula: roots.formula, + ), + ); + return CalculationResult(steps: steps, finalAnswer: roots.finalAnswer); + } else { + // 检查系数是否都是整数 + bool allIntegers = a == a.round() && b == b.round() && c == c.round(); + + if (!allIntegers) { + // 使用公式法 + steps.add( + CalculationStep( + stepNumber: 2, + title: '选择解法', + explanation: '系数包含非整数，我们选择使用公式法。', + formula: r'公式法：$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$', + ), + ); + return _solveQuadraticByFormula(a, b, c, steps); + } else { + // 使用配方法 + steps.add( + CalculationStep( + stepNumber: 2, + title: '选择解法', + explanation: '我们选择使用配方法。', + formula: r'配方法：$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$', + ), + ); + } + } // Step 1: Divide by a if a ≠ 1 String currentEquation; @@ -1453,19 +1565,24 @@ ${b1}y &= ${c1 - a1 * x.toDouble()} final intValue = value.toInt(); if (value == intValue.toDouble()) { // 尝试找到最大的完全平方因子 - int maxK = 1; + int maxSquareRoot = 1; for (int k = 2; k * k <= intValue; k++) { if (intValue % (k * k) == 0) { - maxK = k; + maxSquareRoot = k; } } - if (maxK > 1) { - final remaining = intValue ~/ (maxK * maxK); + if (maxSquareRoot > 1) { + final remaining = intValue ~/ (maxSquareRoot * maxSquareRoot); if (remaining > 1) { - return '$maxK\\sqrt{$remaining}'; + return '$maxSquareRoot\\sqrt{$remaining}'; + } else if (remaining == 1) { + return '$maxSquareRoot'; } } + + // 如果是整数但不是完全平方数，且没有找到 k√m 形式，返回 √value + return '\\sqrt{$intValue}'; } return null; // 无法用简单符号形式表示 @@ -1614,4 +1731,71 @@ ${b1}y &= ${c1 - a1 * x.toDouble()} if (r.denominator == BigInt.one) return r.numerator.toString(); return '\\frac{${r.numerator}}{${r.denominator}}'; } + + /// 使用公式法求解一元二次方程 + CalculationResult _solveQuadraticByFormula( + double a, + double b, + double c, + List steps, + ) { + // Step 3: 计算判别式 + final discriminant = b * b - 4 * a * c; + steps.add( + CalculationStep( + stepNumber: 3, + title: '计算判别式', + explanation: '判别式 Δ = b² - 4ac，用于判断方程根的情况。', + formula: + '\$\$\\Delta = b^2 - 4ac = ${b}^2 - 4 \\cdot ${a} \\cdot ${c} = $discriminant\$\$', + ), + ); + + // Step 4: 应用公式法 + final denominator = 2 * a; + final sqrtDiscriminant = sqrt(discriminant.abs()); + + String formula; + String finalAnswer; + + if (discriminant > 0) { + // 两个实数根 + final x1 = (-b + sqrtDiscriminant) / denominator; + final x2 = (-b - sqrtDiscriminant) / denominator; + + formula = + '\$\$x = \\frac{-b \\pm \\sqrt{\\Delta}}{2a} = \\frac{${-b} \\pm \\sqrt{$discriminant}}{${denominator}}\$\$'; + finalAnswer = '\$\$x_1 = ${x1}, \\quad x_2 = ${x2}\$\$'; + } else if (discriminant == 0) { + // 一个实数根（重根） + final x = -b / denominator; + + formula = '\$\$x = \\frac{-b}{2a} = \\frac{${-b}}{${denominator}}\$\$'; + finalAnswer = '\$\$x = ${x}\$\$'; + } else { + // 两个虚数根 + final realPart = -b / denominator; + final imagPart = sqrtDiscriminant / denominator.abs(); + + formula = + '\$\$x = \\frac{-b \\pm \\sqrt{\\Delta}}{2a} = \\frac{${-b} \\pm \\sqrt{${discriminant}}}{${denominator}}\$\$'; + finalAnswer = + '\$\$x_1 = ${realPart} + ${imagPart}i, \\quad x_2 = ${realPart} - ${imagPart}i\$\$'; + } + + steps.add( + CalculationStep( + stepNumber: 4, + title: '应用公式法', + explanation: discriminant > 0 + ? '判别式大于0，有两个不相等的实数根。' + : discriminant == 0 + ? '判别式等于0，有两个相等的实数根。' + : '判别式小于0，在实数范围内无解，但有虚数根。', + formula: formula, + ), + ); + + return CalculationResult(steps: steps, finalAnswer: finalAnswer); + } }