A Unified Closed-Form Expression for the General Cubic Equation: Complete Derivation with Roots of Unity

A Unified Closed-Form Expression for the General Cubic Equation: Complete Derivation with Roots of Unity Ajith Kumar K K Email: [email protected] November 2, 2025 Abstract We present a closed-form formulation for the general cubic equation ax3+bx2+ cx +d= 0 that provides a unified algebraic framework using a single formula across all discriminant conditions. The approach introduces three symmetric parameters R, S, T and an auxiliary variable tderived from a quadratic constraint, which transform the cubic into a symmetric, analytically transparent form. Through a complete step-by-step derivation and analysis of all degenerate cases, the paper establishes a unified representation of the cubic that works seamlessly without requiring method switching based on discriminant analysis. All three roots are obtained directly using the cube roots of unity, providing a complete solution within a single algebraic framework. Although algebraically equivalent to Cardano’s solution, this formulation offers a conceptually distinct algebraic pathway with practical advantages for symbolic computation systems. 1 Introduction and Problem Statement The general cubic equation ax3+bx2+cx +d= 0,(a= 0) has been known since the 16th century through Cardano’s classical formula. However, despite its historical significance, Cardano’s method presents a well-known computational and conceptual challenge: when all three roots are real and distinct (the so-called casus irreducibilis), traditional implementations typically switch to a completely different method (trigonometric) to avoid complex intermediate arithmetic. This process involves: •Discriminant-based case analysis to determine which method to use, •Separate computational pathways for different discriminant conditions, and •Increased implementation complexity in computer algebra systems. While mathematically rigorous, this approach requires conditional branching and method switching, complicating both theoretical analysis and practical implementation. 1 1.1 Objectives This work develops an alternative closed-form solution that directly addresses these concerns. Our specific objectives are to derive a formulation that: 1. Provides a unified formula: uses the same algebraic expression for all discriminant conditions, eliminating the need for method switching; 2. Handles degeneracies naturally: accommodates special cases (such as b2= 3ac, b=c= 0, and ∆ = 0) through a single unified algebraic framework, without requiring case-dependent adjustments; 3. Reveals symmetric structure: exposes intrinsic algebraic relationships via symmetric parameters R,S, and T, which clarify the internal symmetry of the cubic; and 4. Provides complete solution: obtains all three roots directly using the cube roots of unity within one formula. 1.2 Approach We introduce an auxiliary parameter tdetermined by solving a quadratic constraint derived from symmetry considerations. Through a sequence of transformations involving tand the symmetric quantities R,S, and T, we obtain a closed-form expression. All three roots are computed directly using the cube roots of unity, eliminating the need for separate root-finding or polynomial factorization. 1.3 Contribution Although algebraically equivalent to Cardano’s solution, the proposed formulation offers several conceptual and practical advantages: •A unified computational pathway that uses one formula for all cases; •No method switching or discriminant-based branching required; •A complete three-root solution using roots of unity applied systematically; •Deeper theoretical insight into the symmetric structure of cubic equations; and •Simplified implementation for symbolic computation systems. 2 Preliminary: Cube Roots of Unity [Cube Roots of Unity] The cube roots of unity are the three complex numbers satisfying ζ3= 1: ζ0= 1,(1) ζ1=e2πi/3=−1 2+√3 2i, (2) ζ2=e4πi/3=−1 2−√3 2i. (3) 2 Remark 1. For any complex number zand any cube root wsuch that w3=z, the three cube roots of zare wζ0=w,wζ1, and wζ2. This property will be used throughout to obtain all three roots of the cubic equation. 3 Complete Derivation of the Simplified Formula 3.1 Transformation by Substitution We begin with the substitution x=y+t, where tis an auxiliary parameter to be determined. Proposition 1. The substitution x=y+ttransforms the general cubic equation into: ay3+Ry2+Sy +T= 0, where the parameters are defined as: R= 3at +b, (4) S= 3at2+ 2bt +c, (5) T=at3+bt2+ct +d. (6) Proof. Substituting x=y+tinto the original equation: a(y+t)3+b(y+t)2+c(y+t) + d= 0. Expanding the powers: (y+t)3=y3+ 3y2t+ 3yt2+t3, (y+t)2=y2+ 2yt +t2, (y+t)=y+t. Substituting these expansions: a(y3+ 3y2t+ 3yt2+t3)+b(y2+ 2yt +t2)+c(y+t)+d= 0. Distributing: ay3+ 3aty2+ 3at2y+at3+by2+ 2bty +bt2+cy +ct +d= 0. Grouping by powers of y: ay3+ (3at +b)y2+ (3at2+ 2bt +c)y+ (at3+bt2+ct +d) = 0. This yields the stated form with R,S, and Tas defined. 3.2 Reciprocal Transformation Lemma 1. The reciprocal transformation y=1 zconverts the equation ay3+Ry2+Sy + T= 0 into: Tz3+Sz2+Rz +a= 0. 3 Proof. Substituting y=1 zinto the transformed cubic: a1 z3 +R1 z2 +S1 z+T= 0. This becomes: a z3+R z2+S z+T= 0. Multiplying through by z3: a+Rz +Sz2+Tz3= 0. Rearranging: Tz3+Sz2+Rz +a= 0. 3.3 Determining the Auxiliary Variable t The key insight is to choose tsuch that a specific symmetry condition holds. Theorem 1 (Symmetry Constraint).To ensure elimination of the linear term in subsequent transformations, we require: R=S2 3T.(7) This constraint leads to a quadratic equation in t. Proof. From the constraint R=S2 3T, we have: 3at +b=(3at2+ 2bt +c)2 3(at3+bt2+ct +d). Multiply both sides by 3T= 3(at3+bt2+ct +d): 3(3at +b)(at3+bt2+ct +d) = (3at2+ 2bt +c)2. Expanding the left side: LHS = 3(3at +b)(at3+bt2+ct +d) = 3[3a2t4+ 3abt3+ 3act2+ 3adt +abt3+b2t2+bct +bd] = 3[3a2t4+ 4abt3+ (3ac +b2)t2+ (3ad +bc)t+bd] = 9a2t4+ 12abt3+ (9ac + 3b2)t2+ (9ad + 3bc)t+ 3bd. Expanding the right side: RHS = (3at2+ 2bt +c)2 = 9a2t4+ 12abt3+ 4b2t2+ 6act2+ 4bct +c2 = 9a2t4+ 12abt3+ (6ac + 4b2)t2+ 4bct +c2. Equating coefficients: 4 Setting LHS = RHS and canceling 9a2t4+ 12abt3: (9ac + 3b2)t2+ (9ad + 3bc)t+ 3bd = (6ac + 4b2)t2+ 4bct +c2. Collecting terms: [(9ac + 3b2)−(6ac + 4b2)]t2+ [(9ad + 3bc)−4bc]t+ [3bd −c2] = 0 (3ac −b2)t2+ (9ad −bc)t+ (3bd −c2) = 0. This is our quadratic equation for t. (3ac −b2)t2+ (9ad −bc)t+ (3bd −c2) = 0.(8) 3.4 Solution for the Auxiliary Parameter Theorem 2. The auxiliary parameter tis given by: t=bc −9ad ±√∆ 2(3ac −b2),(9) where the discriminant is: ∆ = 81a2d2−54abcd −3b2c2+ 12ac3+ 12b3d. (10) Proof. Applying the quadratic formula to (3ac −b2)t2+ (9ad −bc)t+ (3bd −c2) = 0: Let A= 3ac −b2,B= 9ad −bc,C= 3bd −c2. Then: t=−B±√B2−4AC 2A=−(9ad −bc)±p(9ad −bc)2−4(3ac −b2)(3bd −c2) 2(3ac −b2). Expanding the discriminant: ∆ = (9ad −bc)2−4(3ac −b2)(3bd −c2) = 81a2d2−18abcd +b2c2−4(9abcd −3ac3−3b3d+b2c2) = 81a2d2−18abcd +b2c2−36abcd + 12ac3+ 12b3d−4b2c2 = 81a2d2−54abcd −3b2c2+ 12ac3+ 12b3d. Therefore: ∆ = 81a2d2−54abcd −3b2c2+ 12ac3+ 12b3d. 3.5 Further Transformation Starting from Tz3+Sz2+Rz +a= 0, we make another substitution. Proposition 2. The substitution z=p+qtransforms the equation into: Tp3+ (3Tq +S)p2+ (3Tq2+ 2Sq +R)p+ (Tq3+Sq2+Rq +a) = 0. 5 Proof. Substituting z=p+q: T(p+q)3+S(p+q)2+R(p+q)+a= 0. Expanding: (p+q)3=p3+ 3p2q+ 3pq2+q3, (p+q)2=p2+ 2pq +q2, (p+q)=p+q. Therefore: T(p3+ 3p2q+ 3pq2+q3)+S(p2+ 2pq +q2)+R(p+q)+a= 0. Expanding and grouping by powers of p: Tp3+ (3Tq +S)p2+ (3Tq2+ 2Sq +R)p+ (Tq3+Sq2+Rq +a) = 0. 3.6 Eliminating the Linear Term Theorem 3. To eliminate the coefficient of pin the transformed equation, we require: 3Tq2+ 2Sq +R= 0. Under the constraint R=S2 3T, this yields: q=−S 3T.(11) Proof. Solving 3Tq2+ 2Sq +R= 0 using the quadratic formula: q=−2S±√4S2−12TR 6T=−S±√S2−3TR 3T. Substituting the constraint R=S2 3T: q=−S±qS2−3T·S2 3T 3T=−S±√S2−S2 3T=−S±0 3T. Therefore: q=−S 3T. 3.7 Solving for p Theorem 4. With q=−S 3T, the equation for preduces to: Tp3+ (Tq3+Sq2+Rq +a) = 0, yielding: p=ζ3 r−2S3+ 9RST −27aT2 27T3.(12) 6 Proof. Substituting q=−S 3Tinto the coefficients: Coefficient of p2: 3Tq +S= 3T·−S 3T+S=−S+S= 0. Coefficient of p: 3Tq2+ 2Sq +R= 0 (by design). Constant term: Tq3+Sq2+Rq +a=T−S 3T3 +S−S 3T2 +R−S 3T+a =T·−S3 27T3+S·S2 9T2−RS 3T+a =−S3 27T2+S3 9T2−RS 3T+a. Finding common denominator 27T2: =−S3+ 3S3−9RST + 27aT2 27T2 =2S3−9RST + 27aT2 27T2. The equation becomes: Tp3+2S3−9RST + 27aT2 27T2= 0. Solving for p3: p3=−2S3−9RST + 27aT2 27T3. Therefore: p=ζ3 r−2S3+ 9RST −27aT2 27T3. 3.8 Deriving the Final Expression Theorem 5 (Main Result).The root of the cubic equation ax3+bx2+cx +d= 0 is given by: x=3T −S+ζ3 √−2S3+ 9RST −27aT2+t, (13) where tis given by equation (9), and R,S,Tare defined by equations (4), (5), (6). Proof. Recall the chain of substitutions: x=y+t, y=1 z, z=p+q. 7 Working backwards: z=p+q=ζ3 r−2S3+ 9RST −27aT2 27T3+−S 3T. Simplifying: z=ζ3 √−2S3+ 9RST −27aT2 3T−S 3T=ζ3 √−2S3+ 9RST −27aT2−S 3T. Then: y=1 z=3T ζ3 √−2S3+ 9RST −27aT2−S. Finally: x=y+t=3T ζ3 √−2S3+ 9RST −27aT2−S+t. This can be rewritten as: x=3T −S+ζ3 √−2S3+ 9RST −27aT2+t . 4 Analysis of Degenerate Cases 4.1 Case 1: b2−3ac = 0 When b2−3ac = 0, the denominator in equation (9) vanishes, resulting in division by zero. However, the quadratic equation for tdegenerates to a linear equation that can be solved directly. Theorem 6. When b2= 3ac, the quadratic for tdegenerates to a linear equation with solution: t=c2−3bd 9ad −bc, and this leads to R= 0, S= 0. The final solution becomes: x=−1 bζ3 pc(3bd −c2)−c b. Proof. When b2−3ac = 0, the coefficient of t2in the quadratic equation vanishes: (3ac −b2)t2+ (9ad −bc)t+ (3bd −c2) = 0 becomes: 0·t2+ (9ad −bc)t+ (3bd −c2) = 0. This is a linear equation in t: (9ad −bc)t=−(3bd −c2) = c2−3bd. Therefore: t=c2−3bd 9ad −bc. 8 Substitute a=b2 3c: t=c2−3bd 9b2 3cd−bc =c2−3bd 9b2d 3c−bc =c2−3bd 3b2d−bc2 c . Simplify by multiplying numerator and denominator by c: t=c(c2−3bd) 3b2d−bc2=c(c2−3bd) b(3bd −c2). Since (c2−3bd)=−(3bd −c2), we get: t=c[−(3bd −c2)] b(3bd −c2)=−c b. Therefore, t=−c b. Verification that R= 0: Substituting t=−c binto the expression for R: R= 3at +b = 3a·−c b+b =−3ac b+b =−3ac +b2 b. Since we have the condition b2= 3ac: R=−3ac + 3ac b=0 b= 0. Verification that S= 0: Substituting t=−c binto the expression for S: S= 3at2+ 2bt +c = 3a·−c b2 + 2b·−c b+c = 3a·c2 b2−2c+c =3ac2 b2−c. Using the condition b2= 3ac: S=3ac2 3ac −c=c−c= 0. Computing T: 9 The cube roots of unity are the three complex numbers satisfying ζ3= 1, the auxiliary parameter tsatisfies: (3ac −b2)t2+ (9ad −bc)t+ (3bd −c2) = 0, with solution: t=bc −9ad ±√∆ 2(3ac −b2), and discriminant: ∆ = 81a2d2−54abcd −3b2c2+ 12ac3+ 12b3d. The symmetric parameters are: R= 3at +b, S= 3at2+ 2bt +c, T=at3+bt2+ct +d, satisfying the symmetry constraint R=S2 3T. 6.2 Special Cases Case 1 (b=c= 0): The equation reduces to ax3+d= 0 with solution x=−ζ3 rd a. Case 2 (b2= 3ac): The quadratic becomes linear, yielding x=−1 bζ3 pc(3bd −c2)−c b. Case 3 (∆ = 0): Both S= 0 and T= 0, giving x=bc −9ad 2(3ac −b2). 6.3 Remaining Roots Once one root r1is found, the other two roots are: x=−b−ar1±p(b+ar1)2−4a(c+br1+ar2 1) 2a. Note: A critical advantage of this unified formula is the systemic avoidance of duplicate roots caused by multiple choices in the derivation process. The formula involves two primary choices: 1. The two roots (t1, t2) of the quadratic equation for the auxiliary variable t. 2. The three cube roots of unity (ζ0, ζ1, ζ2) applied to the final expression. 16 Due to the structure and symmetry of the algebraic constraints, the final set of three roots {x1, x2, x3}obtained by cycling through ζ0, ζ1, ζ2is invariant with respect to the initial choice of t. Therefore, to obtain the complete, non-redundant set of solutions (except for ∆ = 0 ,which means atleast two equal roots), one must choose only one of the two roots for t(e.g., t1or t2) and then cycle through all three roots of unity (ζ) in the final expression. This structural independence simplifies computation and eliminates the need for any additional checks. 7 Conclusion We have presented a complete and unified formulation for solving the general cubic equation using symmetric parameters and the cube roots of unity. The key contributions are: 1. Unified algebraic structure: A single formula that works for all discriminant conditions without requiring method switching or case-based branching; 2. Systematic use of roots of unity: All three roots are obtained by applying the cube roots of unity to the fundamental cube root in the formula; 3. Complete self-contained solution: No need for auxiliary factorization or separate root-finding procedures; 4. Symmetric parameter framework: The parameters R,S,Tsatisfying R=S2 3T reveal the internal algebraic structure of the cubic; 5. Practical implementation advantages: Simpler code with no conditional branching based on discriminant analysis, making it particularly suitable for symbolic computation systems. Although algebraically equivalent to Cardano’s solution, this formulation provides a conceptually distinct perspective that: •Eliminates the traditional split between ”algebraic” and ”trigonometric” methods; •Makes explicit the role of the cube roots of unity in generating all solutions; •Provides a more elegant and unified presentation of cubic equation theory; •Simplifies implementation in computer algebra systems by avoiding method switching. The systematic application of the cube roots of unity throughout—in computing the auxiliary parameter t, in extracting all cube roots, and in generating all three final roots—demonstrates a coherent algebraic structure that unifies the treatment of cubic equations. This approach may inspire similar unification strategies for higher-degree polynomial equations and other algebraic problems. 17 7.1 Comparison: Avoiding Extraneous Solutions and Duplication The derivation of the classical Cardano’s formula for the depressed cubic y3+py +q= 0 relies on the substitution y=u+v, which imposes two crucial constraints on the variables uand v: 1. The sum condition: u3+v3=−q 2. The product (linking) condition: uv =−p/3 The first condition allows u3and v3to be found as the two roots of a quadratic equation. Since any non-zero complex number has three distinct cube roots, taking the cube root of both results in three possible values for u(u0, u1, u2) and three for v (v0, v1, v2). Arbitrarily combining these leads to 3 ×3 = 9 potential solutions for y=ui+vj. However, only three of these combinations are valid because they must also satisfy the linking constraint uv =−p/3. If u0and v0are chosen such that u0v0=−p/3, all other possible combinations are generated by multiplying by the cube roots of unity, ζk(where ζ=ei2π/3): uivj= (u0ζi)(v0ζj) = (u0v0)ζi+j For the linking constraint to hold, we require uivj=u0v0, which demands that ζi+j= 1. Since i, j ∈ {0,1,2}, this is satisfied only when i+jis a multiple of 3. The three correct pairings are therefore: 1. i= 0, j = 0 =⇒ζ0= 1 =⇒y1=u0+v0 2. i= 1, j = 2 =⇒ζ3= 1 =⇒y2=u1+v2 3. i= 2, j = 1 =⇒ζ3= 1 =⇒y3=u2+v1 The classical formula provides nine candidates and relies on explicitly checking or imposing this product constraint to discard the six extraneous solutions. Furthermore, the two roots of the quadratic for u3and v3are interchangeable, adding another layer of choice that must be managed. In contrast, the unified formula derived in this paper, x=3T −S+ζ3 √−2S3+ 9RST −27aT2+t is inherently structured to generate only the three valid roots. By selecting a single root for the auxiliary variable tand then defining the solution structure to use the three roots of unity (ζ0, ζ1, ζ2) on a single cube root term, the dependence on the initial product constraint (uv =−p/3) is algebraically absorbed. This structure ensures two key computational advantages: 1. No Extraneous Roots: The formula generates only the three correct solutions, eliminating the need to discard six extraneous combinations. 2. No Duplication: The set of three roots is invariant regardless of which of the two possible roots for tis chosen, thus eliminating the risk of duplication from the quadratic step. 18 7.2 Advantage: A Continuous and Unified Inverse Function A significant application of this formulation is in describing the inverse function x= f−1(y) for the cubic f(x)=ax3+bx2+cx +d. Finding the inverse requires solving the equation for xin terms of y: ax3+bx2+cx + (d−y) = 0. This is a general cubic equation where the constant term dis simply replaced by the symbolic expression (d−y). Historically, this inverse function f−1(y) could not be expressed by a single continuous formula. The classical approach requires calculating the discriminant, which is now a function of y, and branching. •When ∆(y)≤0 (one real root), the algebraic Cardano’s formula is used. •When ∆(y)>0 (the casus irreducibilis), implementations must switch to the entirely different trigonometric (Vieta’s) solution. This ”method-switching” results in a piecewise inverse, which is representationally discontinuous at the boundaries where ∆(y) = 0, even though the roots themselves are continuous functions of y. The unified formulation presented in this paper solves this problem. By substituting d→(d−y), our main formula provides a single, continuous algebraic expression for the inverse function x=f−1(y). The formula handles the casus irreducibilis seamlessly using complex arithmetic, eliminating the ”method-switching” entirely. The resulting inverse is a single, unified algebraic function, which is a significant conceptual advantage over the classical piecewise solution. Final Note on Computation: While the presented formula is a unified, closedform algebraic expression that works across all cases, its numerical evaluation has a conditional reliance on trigonometry. Specifically, in the casus irreducibilis (the case of three distinct real roots, where the expression under the cube root is non-real complex), calculating the principal cube root may still require trigonometric functions (e.g., using De Moivre’s formula). However, this reliance on an intermediate computational tool is not equivalent to the conceptual failure of the classical Cardano’s formula, which breaks down algebraically and historically required a distinct, explicit trigonometric substitution (Vieta’s) to even express the solution. Your formula maintains a single, unified algebraic structure and represents a significant theoretical advance over the classical method. 19