How does the quadratic formula solve ax² + bx + c = 0?

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) solves any second-degree polynomial. The discriminant Δ = b² - 4ac determines the root nature: if Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one real double root (the parabola touches the x-axis at its vertex); if Δ < 0, there are two complex conjugate roots (x = -b/2a ± i√(4ac - b²)/2a).

What is Cardano’s method for solving cubic equations?

Cardano’s method solves general cubic equations ax³ + bx² + cx + d = 0 by substituting x = t - b/(3a) to eliminate the quadratic term, producing the depressed cubic t³ + pt + q = 0. Depending on the cubic discriminant Δ = (q/2)² + (p/3)³, the equation yields one real root and two complex conjugate roots (Δ > 0) or three distinct real roots via trigonometric substitution (Δ < 0).

How are 2×2 and 3×3 simultaneous systems solved?

The solver applies Cramer’s Rule using determinants. For 2×2 systems: D = a₁b₂ - a₂b₁, D_x = c₁b₂ - c₂b₁, D_y = a₁c₂ - a₂c₁. If D ≠ 0, the unique intersection is x = D_x/D and y = D_y/D. For 3×3 systems, matrix inversion and 3×3 determinants find the intersection vector (x, y, z).

How does the Newton-Raphson non-linear root finder work?

The Newton-Raphson method is an iterative numerical algorithm that finds roots of arbitrary differentiable functions f(x) = 0 using tangent line approximations: x_{k+1} = x_k - f(x_k) / f’(x_k). The solver displays an iteration table showing convergence until tolerance < 10⁻⁸ is reached.

Can this solver handle equations with no solution or infinite solutions?

Yes. For linear equations, 0x = 0 indicates an identity with infinitely many solutions, while 0x = c (c ≠ 0) indicates a contradiction with no solution. For simultaneous systems, D = 0 with zero numerator determinants indicates coincident lines (infinite solutions), while non-zero numerators indicate parallel lines (no solution).

What information does the parabola vertex provide in quadratic equations?

The vertex (h, k) = (-b/(2a), c - b²/(4a)) represents the global minimum (if a > 0) or global maximum (if a < 0) of the quadratic curve, defining the line of symmetry x = h and the range of the quadratic function.

Equation Solver Online — Linear, Quadratic, Cubic & Simultaneous Systems

Q: What is Cardano’s method for solving cubic equations?

Cardano’s method solves general cubic equations ax³ + bx² + cx + d = 0 by substituting x = t - b/(3a) to eliminate the quadratic term, producing the depressed cubic t³ + pt + q = 0. Depending on the cubic discriminant Δ = (q/2)² + (p/3)³, the equation yields one real root and two complex conjugate roots (Δ > 0) or three distinct real roots via trigonometric substitution (Δ < 0).

Q: How are 2×2 and 3×3 simultaneous systems solved?

The solver applies Cramer’s Rule using determinants. For 2×2 systems: D = a₁b₂ - a₂b₁, D_x = c₁b₂ - c₂b₁, D_y = a₁c₂ - a₂c₁. If D ≠ 0, the unique intersection is x = D_x/D and y = D_y/D. For 3×3 systems, matrix inversion and 3×3 determinants find the intersection vector (x, y, z).

Q: How does the Newton-Raphson non-linear root finder work?

The Newton-Raphson method is an iterative numerical algorithm that finds roots of arbitrary differentiable functions f(x) = 0 using tangent line approximations: x_{k+1} = x_k - f(x_k) / f’(x_k). The solver displays an iteration table showing convergence until tolerance < 10⁻⁸ is reached.

Q: Can this solver handle equations with no solution or infinite solutions?

Yes. For linear equations, 0x = 0 indicates an identity with infinitely many solutions, while 0x = c (c ≠ 0) indicates a contradiction with no solution. For simultaneous systems, D = 0 with zero numerator determinants indicates coincident lines (infinite solutions), while non-zero numerators indicate parallel lines (no solution).

Q: What information does the parabola vertex provide in quadratic equations?

The vertex (h, k) = (-b/(2a), c - b²/(4a)) represents the global minimum (if a > 0) or global maximum (if a < 0) of the quadratic curve, defining the line of symmetry x = h and the range of the quadratic function.

Equation Solver Online — Linear, Quadratic, Cubic & Simultaneous Systems

Free online equation solver with step-by-step solutions. Solve linear, quadratic (formula, vertex & factoring), cubic (Cardano reduction), 2×2 & 3×3 simultaneous systems, and non-linear roots with exact fractions & interactive Cartesian graphs.

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Please enable JavaScript in your browser to run live computations, 2D envelope diagrams, and export PDF reports.

How to Use Equation Solver

Select the equation category: Linear (ax + b = c), Quadratic (ax² + bx + c = 0), Cubic (ax³ + bx² + cx + d = 0), 2×2 Simultaneous System, 3×3 Simultaneous System, or Non-Linear (Newton-Raphson f(x) = 0).
Enter the numeric coefficients or function expression in the input fields, or click one of the pre-configured presets (Real Roots, Complex Conjugates, Perfect Squares, Cardano Cubics, Kepler Equations).
Toggle between Decimal and Exact Rational Fraction (a/b) output formats.
Inspect the computed roots, vertex coordinates, discriminant (Δ), and interactive Cartesian graph with root markers.
Review the step-by-step algebraic proof breakdown and export the complete solution as a CSV file.

Algebraic Foundations: From Linear Relations to Higher Polynomials

Algebraic equations model physical trajectories, economic break-even thresholds, circuit node voltages, and optimization bounds. The Fundamental Theorem of Algebra guarantees that every n-th degree polynomial has exactly n complex roots, providing complete predictability in mathematical physics.

Geometric Duality: Visualizing Algebraic Roots on the Coordinate Plane

Every algebraic root corresponds to a geometric x-intercept where the function curve crosses the horizontal axis (f(x) = 0). Linear equations form straight lines, quadratics form parabolas with focal points, and cubics form S-shaped curves with inflection points.

Frequently Asked Questions

How does the quadratic formula solve ax² + bx + c = 0?: The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) solves any second-degree polynomial. The discriminant Δ = b² - 4ac determines the root nature: if Δ > 0, there are two distinct real roots; if Δ = 0, there is exactly one real double root (the parabola touches the x-axis at its vertex); if Δ < 0, there are two complex conjugate roots (x = -b/2a ± i√(4ac - b²)/2a).
What is Cardano’s method for solving cubic equations?: Cardano’s method solves general cubic equations ax³ + bx² + cx + d = 0 by substituting x = t - b/(3a) to eliminate the quadratic term, producing the depressed cubic t³ + pt + q = 0. Depending on the cubic discriminant Δ = (q/2)² + (p/3)³, the equation yields one real root and two complex conjugate roots (Δ > 0) or three distinct real roots via trigonometric substitution (Δ < 0).
How are 2×2 and 3×3 simultaneous systems solved?: The solver applies Cramer’s Rule using determinants. For 2×2 systems: D = a₁b₂ - a₂b₁, D_x = c₁b₂ - c₂b₁, D_y = a₁c₂ - a₂c₁. If D ≠ 0, the unique intersection is x = D_x/D and y = D_y/D. For 3×3 systems, matrix inversion and 3×3 determinants find the intersection vector (x, y, z).
How does the Newton-Raphson non-linear root finder work?: The Newton-Raphson method is an iterative numerical algorithm that finds roots of arbitrary differentiable functions f(x) = 0 using tangent line approximations: x_{k+1} = x_k - f(x_k) / f’(x_k). The solver displays an iteration table showing convergence until tolerance < 10⁻⁸ is reached.
Can this solver handle equations with no solution or infinite solutions?: Yes. For linear equations, 0x = 0 indicates an identity with infinitely many solutions, while 0x = c (c ≠ 0) indicates a contradiction with no solution. For simultaneous systems, D = 0 with zero numerator determinants indicates coincident lines (infinite solutions), while non-zero numerators indicate parallel lines (no solution).
What information does the parabola vertex provide in quadratic equations?: The vertex (h, k) = (-b/(2a), c - b²/(4a)) represents the global minimum (if a > 0) or global maximum (if a < 0) of the quadratic curve, defining the line of symmetry x = h and the range of the quadratic function.