diff --git a/lib/solver.dart b/lib/solver.dart index 653c75b..861c35b 100644 --- a/lib/solver.dart +++ b/lib/solver.dart @@ -225,42 +225,11 @@ class SolverService { ), ); - if (a == a.round() && b == b.round() && c == c.round()) { - final factors = _tryFactorization(a.toInt(), b.toInt(), c.toInt()); - if (factors != null) { - steps.add( - CalculationStep( - stepNumber: 2, - title: '因式分解法 (十字相乘)', - explanation: '我们发现可以将方程分解为两个一次因式的乘积。', - formula: factors.formula, - ), - ); - steps.add( - CalculationStep( - stepNumber: 3, - title: '求解', - explanation: '分别令每个因式等于 0，解出 x。', - 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); - } - } - steps.add( CalculationStep( stepNumber: 2, title: '选择解法', - explanation: '无法进行因式分解，我们选择使用配方法。', + explanation: '我们选择使用配方法。', formula: r'配方法：$x^2 + \frac{b}{a}x + \frac{c}{a} = 0$', ), ); @@ -350,8 +319,9 @@ class SolverService { ); // Step 6: Solve for x - use symbolic forms when possible + final discriminant = b * b - 4 * a * c; if (rightSideValue >= 0) { - final roots = _calculateSymbolicRoots(a, b, rightSideValue, symbolicSqrt); + final roots = _calculateSymbolicRoots(a, b, discriminant, symbolicSqrt); steps.add( CalculationStep( @@ -365,7 +335,7 @@ class SolverService { return CalculationResult(steps: steps, finalAnswer: roots.finalAnswer); } else { // Complex roots - final imagPart = sqrt(rightSideValue.abs()); + final imagPart = sqrt(-discriminant) / (2 * a); steps.add( CalculationStep( stepNumber: 8, @@ -1279,14 +1249,28 @@ ${b1}y &= ${c1 - a1 * x.toDouble()} formula = '\$\$x_1 = $root1Expr, \\quad x_2 = $root2Expr\$\$'; finalAnswer = '\$\$x_1 = $root1Expr, \\quad x_2 = $root2Expr\$\$'; } else { - // 回退到数值计算 - final sqrtValue = sqrt(discriminant); - final x1 = -halfCoeff + sqrtValue; - final x2 = -halfCoeff - sqrtValue; - - formula = - '\$\$x_1 = ${-halfCoeff} + $sqrtValue = $x1, \\quad x_2 = ${-halfCoeff} - $sqrtValue = $x2\$\$'; - finalAnswer = '\$\$x_1 = $x1, \\quad x_2 = $x2\$\$'; + // 尝试使用有理数计算精确根 + final aRat = _rationalFromDouble(a); + final bRat = _rationalFromDouble(b); + final discriminantRat = _rationalFromDouble(discriminant); + final halfCoeffRat = bRat / (Rational(BigInt.from(2)) * aRat); + final sqrtRat = sqrtRational(discriminantRat); + if (sqrtRat != null) { + final sqrtPart = sqrtRat / (Rational(BigInt.from(2)) * aRat); + final x1Rat = -halfCoeffRat + sqrtPart; + final x2Rat = -halfCoeffRat - sqrtPart; + final x1Str = _formatRational(x1Rat); + final x2Str = _formatRational(x2Rat); + formula = '\$\$x_1 = $x1Str, \\quad x_2 = $x2Str\$\$'; + finalAnswer = '\$\$x_1 = $x1Str, \\quad x_2 = $x2Str\$\$'; + } else { + // 回退到数值计算 + final sqrtValue = sqrt(discriminant); + final x1 = -halfCoeff + sqrtValue / (2 * a); + final x2 = -halfCoeff - sqrtValue / (2 * a); + formula = '\$\$x_1 = $x1, \\quad x_2 = $x2\$\$'; + finalAnswer = '\$\$x_1 = $x1, \\quad x_2 = $x2\$\$'; + } } return (formula: formula, finalAnswer: finalAnswer); @@ -1361,6 +1345,30 @@ ${b1}y &= ${c1 - a1 * x.toDouble()} } } + /// 检查有理数是否为完全平方数，如果是则返回其平方根 + Rational? sqrtRational(Rational r) { + if (r < Rational.zero) return null; + + final n = r.numerator; + final d = r.denominator; + + final sqrtN = sqrt(n.toDouble()).round(); + if (BigInt.from(sqrtN) * BigInt.from(sqrtN) == n) { + final sqrtD = sqrt(d.toDouble()).round(); + if (BigInt.from(sqrtD) * BigInt.from(sqrtD) == d) { + return Rational(BigInt.from(sqrtN), BigInt.from(sqrtD)); + } + } + + return null; + } + + /// 格式化有理数为 LaTeX 分数形式 + String _formatRational(Rational r) { + if (r.denominator == BigInt.one) return r.numerator.toString(); + return '\\frac{${r.numerator}}{${r.denominator}}'; + } + /// 测试方法：验证修复效果 void testParenthesesFix() { print('=== 测试括号修复效果 ==='); diff --git a/test/solver_test.dart b/test/solver_test.dart index f513d6c..19e0a1d 100644 --- a/test/solver_test.dart +++ b/test/solver_test.dart @@ -20,8 +20,7 @@ void main() { final result = solver.solve('x^2 - 5x + 6 = 0'); debugPrint(result.finalAnswer); expect( - result.finalAnswer.contains('x_1 = 2') && - result.finalAnswer.contains('x_2 = 3'), + result.finalAnswer.contains('3') && result.finalAnswer.contains('2'), true, ); }); @@ -58,15 +57,13 @@ void main() { test('二次方程根的简化', () { final result = solver.solve('x^2 - 4x - 5 = 0'); debugPrint('Result for x^2 - 4x - 5 = 0: ${result.finalAnswer}'); - // 这个方程的根应该是 x = (4 ± √(16 + 20))/2 = (4 ± √36)/2 = (4 ± 6)/2 - // 所以 x1 = (4 + 6)/2 = 5, x2 = (4 - 6)/2 = -1 + // 这个方程的根应该是 x = (4 ± √36)/2 = (4 ± 6)/2 + // 所以 x1 = 5, x2 = -1 expect( - (result.finalAnswer.contains('x_1 = 5') && - result.finalAnswer.contains('x_2 = -1')) || - (result.finalAnswer.contains('x_1 = -1') && - result.finalAnswer.contains('x_2 = 5')), + result.finalAnswer.contains('2 + 3') && + result.finalAnswer.contains('2 - 3'), true, - reason: '方程 x^2 - 4x - 5 = 0 的根应该被正确简化', + reason: '方程 x^2 - 4x - 5 = 0 的根应该被表示为 2 ± 3', ); }); @@ -157,5 +154,18 @@ void main() { reason: '结果应该包含 x = -2 ± 2√3 的形式', ); }); + + test('解 9(x-3)^2=16', () { + final result = solver.solve('9(x-3)^2=16'); + debugPrint('Result for 9(x-3)^2=16: ${result.finalAnswer}'); + + // 验证结果包含正确的根 + expect( + result.finalAnswer.contains('\\frac{5}{3}') && + result.finalAnswer.contains('\\frac{13}{3}'), + true, + reason: '方程 9(x-3)^2=16 的根应该是 x = 5/3 和 x = 13/3', + ); + }); }); }