A quartic that is biquadratic once depressed is solved

BREAKING-CHANGES.md

@@ -72,6 +72,18 @@ one inside another was answered with the empty set, a claim that it has no solut
 | `"1/(2 + p^(-3)) = 1/3".Solve("p")` | `{ }` | `{ 1, -1/2 + i * 1/2 * sqrt(3), -1/2 + i * -1/2 * sqrt(3) }` |
 | `"1/(1 + x^(-2)) = y".Solve("x")` | `{ }` | `{ ((-1) / (-y) - 1) ^ (-1/2), -((-1) / (-y) - 1) ^ (-1/2) }` |
 
+### A quartic that is biquadratic once depressed is solved
+
+**Silent: a quartic whose depressed form `y^4 + alpha y^2 + gamma`, for `y = x + b/(4a)`, has no cubic term is solved,
+where it was answered `{ }`.** Its roots were shifted back by `-b a / 4` where the shift is `-b/(4a)`, so with a leading
+coefficient other than `1` or `-1` every candidate was off and failed its check, and the empty set claimed there was no
+root. An equation in a square root whose squared form is such a quartic was answered `{ }` with it.
+
+| Input | Was (2.6.0) | Now |
+|---|---|---|
+| `"31x^4 - 62x^3 + 33x^2 - 2x - 1 = 0".ToEntity().Solve("x")` | `{ }` | `{ 1/2 + -sqrt((27/62 + -8/31 * sqrt(2)) / 2), … }`, four real roots |
+| `"x^2 - x - 1 = 4x*sqrt(2x^2 - 4x + 2)".ToEntity().Solve("x")` | `{ }` | `{ 1/2 + -sqrt((27/62 + 8/31 * sqrt(2)) / 2) }` |
+
 ### A quotient of logarithms is differentiated only where both diverge
 
 **Silent: a limit of a quotient of logarithms whose divisor tends to 0 is no longer answered by l'Hopital's rule, which

Sources/.editorconfig

@@ -102,6 +102,9 @@ file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed
 [Tests/UnitTests/Algebra/ExponentialTwoBasesTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 
+[Tests/UnitTests/Algebra/SolveTest/ADepressedBiquadraticQuarticTest.cs]
+file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
+
 [Tests/UnitTests/Algebra/OneByOneMatrixTest.cs]
 file_header_template=\nCopyright (c) 2019-2026 Angouri.\nAngouriMath is licensed under MIT.\nDetails: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.\nWebsite: https://am.angouri.org.\n
 

Sources/AngouriMath/Functions/Continuous/Solvers/EquationSolver/PolynomialSolver.cs

@@ -268,9 +268,14 @@ internal static IEnumerable<Entity> SolveQuartic(Entity a, Entity b, Entity c, E
             var gamma = (-3 * MathS.Pow(b, 4) / (256 * MathS.Pow(a, 4)) + MathS.Sqr(b) * c / (16 * MathS.Pow(a, 3)) - (b * d) / (4 * MathS.Sqr(a)) + e / a)
                 .InnerSimplified;
 
+            // Biquadratic once depressed: y^4 + alpha y^2 + gamma = 0 for y = x + b/(4a), so x is
+            // the root of the quadratic in y^2 shifted back by b/(4a). The shift was written
+            // -b / 4 * a, which is -b a / 4, and for a leading coefficient other than 1 or -1 every
+            // candidate was off and failed its check: 31x^4 - 62x^3 + 33x^2 - 2x - 1 = 0, with four
+            // real roots, was answered { }.
             if (beta.Evaled == 0)
                 return sqrtsOf1.SelectMany(_ => sqrtsOf1,
-                    (s, t) => -b / 4 * a + s * MathS.Sqrt((-alpha + t * MathS.Sqrt(MathS.Sqr(alpha) - 4 * gamma)) / 2));
+                    (s, t) => -b / (4 * a) + s * MathS.Sqrt((-alpha + t * MathS.Sqrt(MathS.Sqr(alpha) - 4 * gamma)) / 2));
 
             var oneThird = Rational.Create(1, 3);
             var P = (-MathS.Sqr(alpha) / 12 - gamma)

Sources/Tests/UnitTests/Algebra/SolveTest/ADepressedBiquadraticQuarticTest.cs

@@ -0,0 +1,63 @@
+//
+// Copyright (c) 2019-2026 Angouri.
+// AngouriMath is licensed under MIT.
+// Details: https://github.com/asc-community/AngouriMath/blob/master/LICENSE.md.
+// Website: https://am.angouri.org.
+//
+
+using System.Linq;
+using AngouriMath.Extensions;
+using Xunit;
+using static AngouriMath.Entity;
+using static AngouriMath.Entity.Set;
+
+namespace AngouriMath.Tests.Algebra
+{
+    /// <summary>
+    /// A quartic whose depressed form <c>y^4 + alpha y^2 + gamma</c>, for <c>y = x + b/(4a)</c>, has
+    /// no cubic term is solved as a quadratic in <c>y^2</c> and shifted back by <c>b/(4a)</c>. The
+    /// shift was written <c>-b / 4 * a</c>, which is <c>-a b / 4</c>, so with a leading coefficient
+    /// other than 1 or -1 every candidate was off and failed its check, and the quartic was
+    /// answered <c>{ }</c>. Found by a cross-check of equations in square roots, whose squared
+    /// forms are such quartics; no issue was filed.
+    /// </summary>
+    [Trait("Area", "Algebra")]
+    public sealed class ADepressedBiquadraticQuarticTest
+    {
+        private static void AssertRoots(string equation, params string[] roots)
+        {
+            var answer = Assert.IsType<FiniteSet>(equation.ToEntity().Solve("x"));
+            Assert.Equal(roots.Length, answer.Count);
+            foreach (var root in roots)
+                Assert.Contains(answer, found => Number.IsZero((found - root.ToEntity()).EvalNumerical()));
+        }
+
+        /// <summary>Irreducible over the rationals, so that no factoring answers it first.</summary>
+        [Fact]
+        public void FourRealRoots()
+            => AssertRoots("31x^4 - 62x^3 + 33x^2 - 2x - 1 = 0",
+                "1/2 + sqrt((27/62 + 8/31 * sqrt(2)) / 2)", "1/2 - sqrt((27/62 + 8/31 * sqrt(2)) / 2)",
+                "1/2 + sqrt((27/62 - 8/31 * sqrt(2)) / 2)", "1/2 - sqrt((27/62 - 8/31 * sqrt(2)) / 2)");
+
+        /// <summary><c>2(x - 1)^4 - 2(x - 1)^2 - 1</c>: two real roots and two that are not.</summary>
+        [Fact]
+        public void TwoRealRootsAndTwoComplex()
+            => AssertRoots("2x^4 - 8x^3 + 10x^2 - 4x - 1 = 0",
+                "1 + sqrt((1 + sqrt(3)) / 2)", "1 - sqrt((1 + sqrt(3)) / 2)",
+                "1 + sqrt((1 - sqrt(3)) / 2)", "1 - sqrt((1 - sqrt(3)) / 2)");
+
+        /// <summary>
+        /// Equations in a square root whose squared form is such a quartic, each answered
+        /// <c>{ }</c>. The first has the one real root its squaring keeps; the second, four
+        /// complex ones.
+        /// </summary>
+        [Fact]
+        public void TheEquationsInARootWhoseSquareIsOne()
+        {
+            AssertRoots("x^2 - x - 1 = 4x * sqrt(2x^2 - 4x + 2)", "(1 - sqrt(1 + 4 / (1 + 4 * sqrt(2)))) / 2");
+            AssertRoots("(2x^2 + 4x - 2) / sqrt(-2x - x^2) = -2",
+                "-1 + sqrt((3 + sqrt(-3)) / 2)", "-1 - sqrt((3 + sqrt(-3)) / 2)",
+                "-1 + sqrt((3 - sqrt(-3)) / 2)", "-1 - sqrt((3 - sqrt(-3)) / 2)");
+        }
+    }
+}