Cubic Equation Solver

About Cubic Equation Solver

The Cubic Equation Solver finds all three roots of any cubic equation in the form ax³ + bx² + cx + d = 0. Enter the four coefficients and get instant results with step-by-step solution using Cardano's method, discriminant analysis, factored form, Vieta's relations, and an interactive graph.

How to Use the Cubic Equation Solver

1. Enter coefficients: Type the values of a, b, c, and d for your cubic equation ax³ + bx² + cx + d = 0. The coefficient a must not be zero.
2. Click "Solve Cubic Equation" to compute all three roots.
3. View the roots: Each root is displayed with a label indicating whether it is real or complex. Real roots appear in green cards, complex roots in blue.
4. Study the step-by-step solution: Follow the full Cardano's method derivation, including the depressed cubic transformation, discriminant calculation, and root extraction.
5. Explore the graph: See the cubic function plotted with real roots marked in green and the inflection point in orange.

What Is a Cubic Equation?

A cubic equation is a polynomial equation of degree three:

\(ax^3 + bx^2 + cx + d = 0\)

where \(a \neq 0\). By the Fundamental Theorem of Algebra, every cubic equation has exactly three roots (counting multiplicity), which may be real or complex numbers.

Cardano's Formula

Published in 1545 by Gerolamo Cardano (though discovered earlier by Scipione del Ferro and Niccolo Tartaglia), this method works by:

1. Depressing the cubic: Substituting \(x = t - \frac{b}{3a}\) eliminates the \(x^2\) term, yielding \(t^3 + pt + q = 0\)
2. Computing p and q: \(p = \frac{3ac - b^2}{3a^2}\), \(q = \frac{2b^3 - 9abc + 27a^2d}{27a^3}\)
3. Applying the formula: \(t = \sqrt[3]{-\frac{q}{2} + \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\frac{q^2}{4} + \frac{p^3}{27}}}\)

The Discriminant

The discriminant \(\Delta = -4p^3 - 27q^2\) determines the nature of the roots:

• \(\Delta > 0\): Three distinct real roots (uses the trigonometric/Viete method)
• \(\Delta = 0\): At least two equal roots (a repeated root exists)
• \(\Delta < 0\): One real root and two complex conjugate roots

Vieta's Formulas for Cubic Equations

If \(x_1, x_2, x_3\) are the three roots of \(ax^3 + bx^2 + cx + d = 0\), then:

• \(x_1 + x_2 + x_3 = -\frac{b}{a}\) (sum of roots)
• \(x_1 x_2 + x_1 x_3 + x_2 x_3 = \frac{c}{a}\) (sum of products of pairs)
• \(x_1 x_2 x_3 = -\frac{d}{a}\) (product of roots)

Special Cases

• Depressed cubic (\(b = 0\)): \(x^3 + cx + d = 0\) — already in simplified form
• Pure cubic (\(b = c = 0\)): \(ax^3 + d = 0\) — root is \(x = \sqrt[3]{-d/a}\)
• Sum/difference of cubes: \(x^3 \pm k^3 = (x \pm k)(x^2 \mp kx + k^2)\)

Applications of Cubic Equations

• Engineering: Beam deflection, stress analysis, and control systems
• Physics: Barker's equation for parabolic orbits, equations of state (van der Waals)
• Economics: Cost optimization, supply-demand equilibrium models
• Computer graphics: Bezier curves, spline interpolation
• Chemistry: pH calculations involving weak acids/bases

FAQ

What is a cubic equation?

A cubic equation is a polynomial equation of degree 3, written in the form ax³ + bx² + cx + d = 0, where a is not zero. Every cubic equation has exactly three roots, which may be real or complex numbers.

How does Cardano's formula work?

Cardano's formula solves cubic equations by first reducing the equation to a depressed cubic (without the x² term) through a substitution, then applying a formula involving cube roots. The depressed cubic t³ + pt + q = 0 is solved using t = cube_root(-q/2 + sqrt(q²/4 + p³/27)) + cube_root(-q/2 - sqrt(q²/4 + p³/27)).

What does the discriminant of a cubic equation tell you?

The discriminant determines the nature of the roots. If positive, there are three distinct real roots. If zero, there are repeated roots. If negative, there is one real root and two complex conjugate roots.

Can a cubic equation have all complex roots?

No. Every cubic equation with real coefficients has at least one real root. Complex roots always come in conjugate pairs, so a cubic has either three real roots or one real root and two complex conjugate roots.

What is Vieta's formulas for cubic equations?

Vieta's formulas relate the roots to the coefficients. For ax³ + bx² + cx + d = 0 with roots r1, r2, r3: the sum of roots equals -b/a, the sum of products of pairs equals c/a, and the product of all roots equals -d/a.

How can I check whether a cubic root is correct?

Substitute the root into the original polynomial and check that the result is zero. For example, for x^3 - 6x^2 + 11x - 6, x=1 gives 1 - 6 + 11 - 6 = 0. You can also divide the polynomial by (x - 1); if the remainder is zero, 1 is a root. For complex roots, substitution uses complex arithmetic, and rounding may leave a tiny residual.

What does it mean when a cubic has a repeated root?

A repeated root occurs more than once among the cubic’s three roots when counted with multiplicity. For example, (x - 1)^2(x + 2) has roots 1, 1, and -2, so x=1 is repeated. The graph touches the x-axis at x=1 and turns back rather than crossing there. A repeated root still counts toward the total of three roots.

Why does a cubic graph sometimes show only one x-intercept when it has three roots?

The other roots may be complex, and a real graph only displays real x-values. For example, x^3 + 1 = 0 has one real root, -1, and two complex roots. A cubic always has three roots when counted over the complex numbers, including multiplicity, but its graph can cross the real x-axis once or three times. Repeated real roots can touch the axis without crossing it.

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