Quartic Equation Solver

Solve Quartic Equations

Quartic Equation Formula

Formula

\[ ax^4 + bx^3 + cx^2 + dx + e = 0, \quad a \neq 0 \]

Where:

\(a, b, c, d, e\) — real coefficients of the polynomial
\(a\) — leading coefficient (must not be zero, otherwise the equation is not quartic)
\(x\) — the unknown to solve for

This is the general form of a fourth-degree polynomial equation. Solving it analytically uses Ferrari's method: the equation is first reduced to a depressed quartic (no \(x^3\) term), then a resolvent cubic is solved, and the four roots are recovered from square roots of expressions involving that cubic's solution. This calculator carries out the full procedure automatically.

Quartic Equation – Calculation Example

Example: solve \(x^4 - 3x^3 + 3x^2 - x = 0\), which corresponds to the default values \(a = 1\), \(b = -3\), \(c = 3\), \(d = -1\), \(e = 0\).

\[ x^4 - 3x^3 + 3x^2 - x = x(x^3 - 3x^2 + 3x - 1) = x(x - 1)^3 = 0 \]

Result: \(x_1 = 0\) and \(x_2 = x_3 = x_4 = 1\). The equation has a simple root at 0 and a triple root at 1 — the same four values the calculator returns.

How to Use the Calculator

Enter the coefficient a of the \(x^4\) term. It must be non-zero.
Enter the coefficient b of the \(x^3\) term.
Enter the coefficient c of the \(x^2\) term.
Enter the coefficient d of the \(x\) term.
Enter the constant term e. Use 0 if the equation has no constant.
Read the four roots \(x_1\) through \(x_4\). Complex roots appear in the form \(p + qi\) or \(p - qi\).

Use negative values for coefficients with minus signs — for example, for \(x^4 - 3x^3\) enter a = 1 and b = −3. The equation is normalized internally, so any non-zero leading coefficient works, including fractions such as 0.5 (entered as 0.5 or with decimal steps).

What Is a Quartic Equation?

A quartic equation is a polynomial equation of degree four. By the fundamental theorem of algebra it always has exactly four roots in the complex plane, counted with multiplicity. Depending on the coefficients, those roots may be all real, all complex, or a mix of the two. Because the coefficients entered here are real, any complex roots always come in conjugate pairs \(p \pm qi\).

The quartic is the highest degree for which a general solution in radicals exists. Ferrari's method, discovered in the 16th century, reduces the problem to a cubic equation (the resolvent) and then finishes with square roots. This is why the calculator can return exact algebraic values rather than mere numerical guesses. The trade-off is that the full closed-form expression is extremely long, which is why automated tools like this one are used in practice.

Interpreting the results is straightforward: real roots are the points where the polynomial curve crosses the x-axis, while complex roots indicate the curve does not touch the axis at those locations. Repeated roots (like the triple root in the example above) mean the curve is tangent to the axis at that point. If you only need a quick sense of how many real roots to expect before solving, Descartes' Rule of Signs is a handy check.

Useful relationships between the roots and the coefficients (Vieta's formulas) let you verify the answers:

\(x_1 + x_2 + x_3 + x_4 = -b/a\)
\(x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4 = -d/a\)
\(x_1 x_2 x_3 x_4 = e/a\)

Assumptions, Limitations and Tips

Coefficient a must not be zero. If you enter a = 0, the calculator reports an error because the equation is no longer quartic. If \(a = 0\) but \(b \neq 0\), it is actually a cubic; solve it with the Cubic Equation calculator instead. If both a and b are zero, use the Quadratic Equation calculator.
Roots are rounded to three decimal places, so exact values like \(\sqrt{2}\) appear as 1.414. Near-identical roots may look the same after rounding even when they differ slightly.
Numerical precision: for extremely large or tiny coefficients the intermediate resolvent cubic can lose precision. Rescaling the equation (dividing all coefficients by a power of 10) improves stability.
Complex output is normal: if the calculator returns pairs such as \(2 + 3i\) and \(2 - 3i\), the polynomial has no real roots at those positions. Read more about such values with the Complex Root calculator.
Multiplicity counts: four roots are always reported, so a double root appears twice. In the example above, \(x = 1\) is listed three times because it is a triple root.

Frequently Asked Questions

How do I solve a quartic equation by hand?

Start by factoring out any common power of \(x\), then test simple candidates using the rational root theorem — many textbook quartics have roots like 1, −2, or 0.5, and each found root lets you divide the polynomial down to a cubic or quadratic. If no easy root exists, the full hand solution requires Ferrari's method: substitute \(x = t - b/(4a)\) to remove the cubic term, then solve a resolvent cubic and take square roots. The calculator automates exactly this procedure.

How can I tell how many real roots my quartic has before solving?

A quick way is Descartes' Rule of Signs: count sign changes in the coefficient sequence \(a, b, c, d, e\) for positive roots, and in \(a, -b, c, -d, e\) for negative ones. The actual count equals the sign changes or fewer by an even number. Also useful: a quartic with real coefficients always has 0, 2, or 4 real roots (counted with multiplicity) — never exactly 1 or 3 distinct ones, because complex roots come in conjugate pairs.

What do complex roots like 2 + 3i mean physically?

A complex root \(p + qi\) with \(q \neq 0\) means the polynomial curve never touches the x-axis at that location — instead of crossing, the curve makes a "dip" or "hump" above or below the axis near \(x = p\). The real part \(p\) tells you where that dip sits, and the imaginary part \(q\) relates to how shallow or steep it is. In engineering, complex roots indicate oscillating or vibrating behavior.

Can a quartic equation have fewer than four different roots?

Yes. Every quartic has four roots counted with multiplicity, but some may coincide. For example, \(x^4 = 0\) has the single root \(x = 0\) with multiplicity four, and the calculator lists it four times. A repeated root means the curve is tangent to the x-axis there rather than cutting straight through it — the graph just touches the axis and turns around.

Why does the calculator show an error when a = 0?

With \(a = 0\) the highest power vanishes, so the equation is no longer fourth-degree and Ferrari's method does not apply. If \(b \neq 0\), you actually have a cubic; if \(b = 0\) too, it is a quadratic or even linear equation. Use the dedicated Cubic or Quadratic Equation calculators for those cases — they are faster and numerically more stable for lower degrees.

My hand-check of a root doesn't give exactly zero — is the answer wrong?

Probably not. The calculator rounds every root to three decimal places, so substituting a rounded value like 1.414 into the polynomial gives a small residue such as 0.0002 instead of exactly 0. That is expected rounding behavior, not an error. For a stronger check, use Vieta's formulas: the four roots must satisfy \(x_1 + x_2 + x_3 + x_4 = -b/a\) and \(x_1 x_2 x_3 x_4 = e/a\) within rounding tolerance.

What are the most common mistakes when entering coefficients?

The biggest one is forgetting negative signs — for \(x^4 - 5x^2 + 4\) you must enter c = −5, not 5, and leave b and d as 0. Also remember to include zero coefficients for missing terms instead of skipping fields, and watch out for equations not yet in standard form: \(2x^4 = x^3\) must be rewritten as \(2x^4 - x^3 = 0\) (a = 2, b = −1, c = d = e = 0) before entering the values.