Symbolic Algebra & Polynomial Equation Solver
Solve linear, quadratic (ax^2+bx+c=0), cubic, polynomial, and multi-variable systems of equations symbolically and numerically with step-by-step discriminant analysis, factoring, and radical derivations.
# Calculation Pipeline (Chain of Thought)
- Step 1: Canonical Standard Form Reduction: Ingest arbitrary equation string (e.g.
3x^2 - 5 = 2x + 7) and algebraically normalize to canonical zero form: f(x) = 3x^2 - 2x - 12 = 0. - Step 2: Discriminant Analysis (Delta): For quadratic
ax^2 + bx + c = 0, calculate Delta = b^2 - 4ac. If Delta > 0: two real distinct roots; if Delta = 0: one real repeated root; if Delta < 0: two complex conjugate roots x = alpha pm eta i. - Step 3: Closed-Form Formula & Factoring: Execute quadratic formula:
x = rac{-b pm sqrt{b^2 - 4ac}}{2a}. For linear equations ax + b = 0, directly solve x = - rac{b}{a}. - Step 4: Multi-Variable Systems (Linear Elimination): For systems of equations
Ax = B, construct augmented matrix [A | B], compute determinant det(A), and apply Gaussian elimination with back-substitution to solve variable vector (x, y, z).
Decision Rule: Fundamental Theorem of Algebra: Every single-variable polynomial of degree n with complex coefficients has exactly n complex roots (counting algebraic multiplicity). A quadratic (degree 2) always possesses exactly 2 roots.
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| SYMBOLIC ALGEBRAIC EQUATION SOLVER |
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| Equation: 2x^2 - 4x - 6 = 0 ---> a = 2, b = -4, c = -6 |
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| Step 1: Compute Discriminant (Δ): |
| Δ = b^2 - 4ac = (-4)^2 - 4(2)(-6) = 16 - (-48) = 16 + 48 = 64 |
| Since Δ = 64 > 0, there are 2 real, rational distinct roots! |
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| Step 2: Apply Quadratic Formula: |
| x = [ -(-4) ± sqrt(64) ] / [ 2 * 2 ] |
| x = [ 4 ± 8 ] / 4 |
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| Root 1 (x1): (4 + 8) / 4 = 12 / 4 = 3.0 |
| Root 2 (x2): (4 - 8) / 4 = -4 / 4 = -1.0 |
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| Factored Form: 2(x - 3)(x + 1) = 0 |
| Vertex (h, k): h = -b/(2a) = 4/4 = 1.0; k = 2(1)^2 - 4(1) - 6 = -8.0 |
| Vertex Form: 2(x - 1.0)^2 - 8.0 = 0 |
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Frequently Asked Questions
What does the discriminant (b^2 - 4ac) reveal about a quadratic equation?
The discriminant indicates the nature and count of solutions without full evaluation: 1) If Δ > 0, the parabola intersects the x-axis twice (two real distinct roots); 2) If Δ = 0, the vertex touches the x-axis tangentially (one real double root); 3) If Δ < 0, the parabola never crosses the x-axis (two complex conjugate roots containing imaginary unit i = sqrt(-1)).
What is an extraneous solution and how do they arise?
An extraneous solution is a mathematical root that emerges during algebraic manipulation (such as squaring both sides of an equation or multiplying by a variable denominator) that is not a valid solution of the original equation. For example, solving sqrt(x) = -3 by squaring yields x = 9, but sqrt(9) = +3 != -3. Always substitute roots back into original expressions.
How does Gaussian elimination solve systems of linear equations?
Gaussian elimination converts a system's augmented matrix into row-echelon form using three elementary row operations: swapping rows, multiplying a row by a non-zero scalar, and adding a scalar multiple of one row to another. Once in upper-triangular form, back-substitution yields exact values for all unknown variables.
Can all polynomial equations be solved using a radical formula like the quadratic equation?
No. The Abel-Ruffini Theorem (1799/1824) proved that there is no general algebraic solution in radicals for polynomial equations of degree 5 (quintic) or higher. While linear, quadratic, cubic (Cardano's formula), and quartic (Ferrari's formula) equations possess closed-form radical solutions, degree 5+ polynomials require numerical approximation algorithms like Newton-Raphson.
Algebraic Proofs & Solution Checklists
- Canonical Zero Equation Form: Always balance your equation so one side equals zero (f(x) = 0) before identifying coefficients a, b, and c to prevent sign reversal errors.
- Zero-Product Property: If A * B = 0, then either A = 0 or B = 0 (or both). This fundamental algebraic law enables solving polynomials by factoring into linear terms.
- Complex Conjugate Root Pairs: For polynomials with real coefficients, complex roots always appear in conjugate pairs (a + bi and a - bi).
- Determinant Invertibility Check: In a system of linear equations Ax = B, a unique solution exists if and only if the determinant det(A) != 0. If det(A) = 0, the system has either zero or infinite solutions.
- Vieta's Formulas for Root Verification: Quickly verify roots using Vieta's relations: for quadratic ax^2+bx+c=0, the sum of roots x1 + x2 = -b/a and the product of roots x1 * x2 = c/a.