Cubic Equation Solver

Cubic Equation

Cubic Equation Formula

Formula

\[ ax^3 + bx^2 + cx + d = 0, \quad a \neq 0 \] \[ \text{Substitute } x = t - \frac{b}{3a} \text{ to get the depressed cubic: } t^3 + pt + q = 0 \] \[ p = \frac{3ac - b^2}{3a^2}, \qquad q = \frac{2b^3 - 9abc + 27a^2 d}{27a^3} \]

Where:

\(a, b, c, d\) — the coefficients of the cubic equation
\(p, q\) — the coefficients of the depressed cubic after removing the quadratic term

From here, Cardano's method applies. The key quantity is the discriminant \(\Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3\):

If \(\Delta > 0\), there is one real root, given by
\( t = \sqrt[3]{-\frac{q}{2} + \sqrt{\Delta}} + \sqrt[3]{-\frac{q}{2} - \sqrt{\Delta}} \)
If \(\Delta = 0\), all roots are real and at least two coincide: \( t_1 = 2\sqrt[3]{-\frac{q}{2}} \), \( t_2 = t_3 = -\sqrt[3]{-\frac{q}{2}} \)
If \(\Delta < 0\), there are three distinct real roots, found with the trigonometric form \( t_k = 2\sqrt{-\frac{p}{3}} \cos\left(\frac{\theta + 2\pi k}{3}\right) \), where \( \cos\theta = \frac{3q}{2p}\sqrt{-\frac{3}{p}} \), \( k = 0, 1, 2 \)

Each root of the original equation is recovered by back-substitution: \( x = t - \frac{b}{3a} \).

Cubic Equation – Calculation Example

Example: solve \( x^3 + 8x^2 + 16x = 0 \) (coefficients \( a = 1 \), \( b = 8 \), \( c = 16 \), \( d = 0 \) — the calculator's default values).

\[ p = \frac{3 \cdot 1 \cdot 16 - 8^2}{3 \cdot 1^2} = \frac{48 - 64}{3} = -\frac{16}{3}, \qquad q = \frac{2 \cdot 8^3 - 9 \cdot 1 \cdot 8 \cdot 16 + 27 \cdot 1^2 \cdot 0}{27 \cdot 1^3} = \frac{64}{27} \]

\[ \Delta = \left(\frac{q}{2}\right)^2 + \left(\frac{p}{3}\right)^3 = \left(\frac{32}{27}\right)^2 + \left(-\frac{16}{9}\right)^3 = \frac{1024}{729} - \frac{4096}{729} \cdot \frac{1}{...} = 0 \]

Since \( \Delta = 0 \), the equation has a repeated root. With \( \sqrt[3]{-\frac{q}{2}} = \sqrt[3]{-\frac{32}{27}} = -\frac{4}{3} \):

\[ t_1 = 2\left(-\frac{4}{3}\right) = -\frac{8}{3}, \qquad t_2 = t_3 = -\left(-\frac{4}{3}\right) = \frac{4}{3} \]

Back-substituting \( x = t - \frac{8}{3} \):

\[ x_1 = -\frac{8}{3} - \frac{8}{3} = 0, \qquad x_2 = x_3 = \frac{4}{3} - \frac{8}{3} = -4 \]

Result: \( x_1 = 0 \), \( x_2 = -4 \), \( x_3 = -4 \). You can verify this by factoring: \( x^3 + 8x^2 + 16x = x(x+4)^2 \).

How to Use the Calculator

Enter the leading Coefficient a (the number multiplying \( x^3 \)). It must not be 0 — otherwise the equation is not cubic, and the calculator shows an error.
Enter Coefficient b, the multiplier of \( x^2 \).
Enter Coefficient c, the multiplier of \( x \).
Enter Coefficient d, the constant term. Use 0 if the equation has no constant (as in \( x^3 + 8x^2 + 16x = 0 \)).
Read the results: Root x₁, Root x₂ and Root x₃. If the equation has only one real solution, the other two fields display “Complex root”, since the remaining two roots are complex conjugates.

What Is a Cubic Equation?

A cubic equation is any polynomial equation of degree three, written in the standard form \( ax^3 + bx^2 + cx + d = 0 \) with \( a \neq 0 \). Every cubic equation with real coefficients has at least one real root and at most three. Unlike a quadratic equation, which can be solved with a single compact formula, a cubic requires either factoring, Cardano's method, or a numerical approach — which is why a solver tool saves so much time.

The calculator works by first eliminating the \( x^2 \) term. The substitution \( x = t - \frac{b}{3a} \) turns any cubic into a depressed cubic \( t^3 + pt + q = 0 \), a simpler form known since the 16th century. Cardano's formula then computes a discriminant-like quantity \( \Delta \) that tells you the nature of the roots before they are even computed: one real plus two complex roots when \( \Delta > 0 \), all real with a repeated root when \( \Delta = 0 \), and three distinct real roots when \( \Delta < 0 \).

Interpreting the output is straightforward. If all three roots are numbers, the graph of \( y = ax^3 + bx^2 + cx + d \) crosses the x-axis at those points (touching it at a repeated root). If two fields say “Complex root”, the graph crosses the x-axis only once, at the single real root shown. The two hidden roots are a complex conjugate pair; you can explore them further with the complex number calculator.

Cubic equations appear everywhere in applied mathematics: finding intersection points in geometry, computing eigenvalues in linear algebra, modeling volume problems in physics, and analyzing stability in engineering and economics.

Assumptions, Limitations and Tips

Coefficient a must be nonzero. If \( a = 0 \), the equation is actually quadratic (or linear) — use the quadratic equation solver instead.
Complex roots are flagged, not computed. The calculator reports real roots numerically and shows “Complex root” for the non-real pair. This keeps the output clean for everyday use.
Rounding to 5 decimal places. Results are rounded, so a root like \( \sqrt[3]{2} \) appears as 1.25992. For most purposes this is more than enough precision; tiny discrepancies may appear if you substitute a rounded root back into the equation.
Numerical edge cases. Very large or very small coefficients can cause floating-point round-off, so a discriminant that is mathematically exactly zero may compute as a very small positive or negative number. In such cases the displayed roots may differ in the last decimals.
Check with Descartes' rule. Before solving, you can estimate how many positive and negative real roots exist using the Descartes' Rule of Signs calculator, then use the result as a sanity check on the solver's output. Hand in hand with that, the cubic root calculator is useful when you only need \( \sqrt[3]{n} \) for a single number.
Simple factoring first. Always check whether \( x = 0 \) is a root (i.e., \( d = 0 \)) or whether an obvious integer like 1, −1, 2 or −2 satisfies the equation — factoring out one root reduces the cubic to a quadratic you can solve mentally.

Frequently Asked Questions

How do I solve a cubic equation by hand without Cardano's formula?

First try the rational root theorem: any rational root \( \frac{m}{n} \) has \( m \) dividing the constant \( d \) and \( n \) dividing the leading coefficient \( a \). Test candidates like ±1, ±2, ±d by substitution. Once one root \( x_1 \) is found, divide the polynomial by \( (x - x_1) \) (long division or synthetic division) to get a quadratic, then apply the quadratic formula. This covers most textbook cubics.

How many real roots can a cubic equation have?

Always at least one and at most three. The page's quantity \( \Delta \) decides: \( \Delta > 0 \) gives one real root plus a complex conjugate pair, \( \Delta = 0 \) gives three real roots with at least two equal, and \( \Delta < 0 \) gives three distinct real roots. You can also check quickly with the graph: an odd-degree polynomial with real coefficients must cross the x-axis at least once.

How can I verify the roots the calculator gives me?

Substitute each root back into \( ax^3 + bx^2 + cx + d \) — the result should be 0 (or very close, since values are rounded to 5 decimals). You can also use Vieta's formulas as a cross-check: the sum of the roots equals \( -\frac{b}{a} \) and their product equals \( -\frac{d}{a} \). For the example \( x(x+4)^2 \), the sum is \( 0 - 4 - 4 = -8 = -\frac{8}{1} \).

Why does the calculator say “Complex root” instead of showing a number?

When \( \Delta > 0 \) the two remaining roots are complex conjugates \( u \pm vi \). The calculator only reports numerical real roots, so it flags the non-real pair rather than printing complex values. If you need them, note that the real part \( u \) equals \( -\frac{b}{3a} \) minus half the real root, and use the quadratic factor that remains after dividing out the real root.

What does a repeated root mean geometrically?

A repeated (double) root is a point where the curve \( y = ax^3 + bx^2 + cx + d \) touches the x-axis without crossing it, like a parabola's vertex on the axis. In the default example, \( x(x+4)^2 = 0 \), the graph crosses at \( x = 0 \) and merely touches the axis at \( x = -4 \). A triple root (all three equal) looks like an inflection point sitting exactly on the axis.

Can I enter fractions or decimals as coefficients?

The input fields accept decimals, including negative ones (e.g., 0.5 or -2.75). For fractional coefficients like \( \frac{1}{2}x^3 - 3x + 4 \), either convert to decimals (a = 0.5) or multiply the whole equation by the common denominator first — multiplying every term by the same nonzero number does not change the roots.

What are common mistakes when using a cubic equation solver?

The most frequent errors are entering terms in the wrong order (remember: a multiplies \( x^3 \), d is the plain constant), forgetting a zero coefficient (in \( x^3 - 8 = 0 \) you must enter b = 0 and c = 0), and dropping minus signs when copying from paper. Also note that \( x^3 + 5x^2 \) is not complete — missing terms are not automatically assumed to be zero unless you type 0.