Normalized coefficients
- $p$: —
- $q$: —
- $r$: —
Roots
- $x_1$: —
- $x_2$: —
- $x_3$: —
- $x_4$: —
Step-by-step
Explanation:
Depressed Quartic Equation Solver
This depressed quartic equation solver is designed for equations in the form $a x^4 + c x^2 + d x + e = 0$, where the cubic term is absent. It works as a practical quartic root finder for both real and complex solutions, showing normalized coefficients, computed roots, and residual-based accuracy checks so you can verify solution quality. For related polynomial and equation-solving methods, explore Algebra.
Solve quartic equations in depressed form $a x^4 + c x^2 + d x + e = 0$ and inspect real/complex roots.
Coefficient $a$ must be non-zero.
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What This Quartic Calculator Solves
This quartic calculator solves depressed quartic polynomials of degree four: $a x^4 + c x^2 + d x + e = 0$ with $a \neq 0$. It returns all four roots and classifies behavior through real-root count and complex-conjugate output when applicable. This is useful when a closed-form factorization is not obvious and you need dependable numeric roots quickly.
Step-by-Step Normalization and Root Computation
The solver first converts to monic form: $x^4 + p x^2 + q x + r = 0$, where $p=c/a$, $q=d/a$, and $r=e/a$. It then applies an iterative numerical method to compute all roots and reports residual values $|f(x_i)|$ so you can assess numerical stability. This combination supports both fast solving and transparent validation.
Interpreting Real vs Complex Quartic Roots
Quartic equations may produce four real roots, two real plus one complex-conjugate pair, or two complex-conjugate pairs. Complex results are expected in many polynomial systems and should be interpreted as valid solutions over the complex field. When residuals are very small, the computed roots are numerically consistent with the original polynomial.
If you are working with lower-degree cases, use the Quadratic Equation Solver or Completing the Square Calculator for direct quadratic workflows.
Worked Examples (Quartic Root Finder)
Example 1: Four Real Roots
Use this baseline case when the equation factors cleanly over reals. It is the best quick check for whether the solver, root ordering, and residual logic are all behaving correctly.
Given coefficients: $$a=1,\; c=-5,\; d=0,\; e=4$$ The equation is: $$x^4 - 5x^2 + 4 = 0$$
Normalize to monic form: $$x^4 + px^2 + qx + r = 0$$ with $$p=\frac{c}{a}=-5,\quad q=\frac{d}{a}=0,\quad r=\frac{e}{a}=4$$
Factorization: $$(x^2-1)(x^2-4)=0$$ So the real roots are: $$x=\pm1,\;\pm2$$
Residual check mirrors the calculator output and confirms numeric consistency: $$|f(-2)|\approx 0,\quad |f(-1)|\approx 0,\quad |f(1)|\approx 0,\quad |f(2)|\approx 0$$
Root interpretation: four real roots with no complex component. Accuracy check: each reported root should satisfy $$|f(x_i)| \approx 0$$. This case is ideal for confirming exact-root behavior.
Example 2: Mixed Coefficients With Complex Output
Use this when the quartic is not easily factorable and you expect complex-conjugate structure. It demonstrates practical quartic root-finder behavior beyond textbook integer roots.
Given coefficients: $$a=1,\; c=2,\; d=-3,\; e=1$$ The equation is: $$x^4 + 2x^2 - 3x + 1 = 0$$
Normalize: $$p=\frac{c}{a}=2,\quad q=\frac{d}{a}=-3,\quad r=\frac{e}{a}=1$$ then solve numerically for all four roots.
Typical root pattern from the solver is two complex-conjugate pairs. One representative output set is: $$x_1\approx -0.570696-1.62477i,\quad x_2\approx -0.570696+1.62477i$$ $$x_3\approx 0.570696-0.10728i,\quad x_4\approx 0.570696+0.10728i$$
The quality check is not just "root type"; it is also residual consistency: $$|f(x_1)|\approx 9.93\times10^{-16},\quad |f(x_2)|\approx 9.93\times10^{-16}$$ $$|f(x_3)|\approx 0,\quad |f(x_4)|\approx 0$$
Expected interpretation: at least one complex-conjugate root pair and often no exact integer roots. Validate the solution with residual checks: $$|f(x_i)| \approx 0$$ within floating-point tolerance.
Example 3: Uneven Coefficients, Mixed Root Behavior
This example stresses numerical behavior with uneven coefficients and sign changes. It is useful when modeling non-symmetric quartics where analytic factoring is unlikely.
Given coefficients: $$a=2,\; c=-7,\; d=5,\; e=-9$$ The equation is: $$2x^4 - 7x^2 + 5x - 9 = 0$$
Normalize to monic form: $$x^4 + px^2 + qx + r = 0$$ $$p=-3.5,\quad q=2.5,\quad r=-4.5$$ Then compute all four roots numerically and inspect real-root count plus root type.
In practice, this kind of input often produces a mixed root set: one or more real roots plus a complex-conjugate pair. The solver's real/complex badges help classify that immediately, while the full root list shows the exact distribution.
Expected interpretation: mixed real/complex behavior is possible. Do not judge quality by visual root shape alone; use residuals for final confidence.
Validation tip: $$|f(x_i)| \approx 0$$ should hold for every reported root within numerical tolerance.
Example 4: Complex-Heavy Non-Factorable Quartic
This is a deliberate complex-heavy scenario used to verify conjugate-pair symmetry and stable convergence even when the equation resists simple factorization.
Given coefficients: $$a=1,\; c=1,\; d=1,\; e=1$$ The equation is: $$x^4 + x^2 + x + 1 = 0$$
Normalize: $$p=1,\quad q=1,\quad r=1$$ and compute roots numerically. Expect complex-conjugate roots to dominate.
This scenario is especially useful for debugging interpretation rules: root type should be complex, real-root count should be zero, and residuals should remain very small despite non-factorable structure.
Accuracy confirmation: use max residual plus each $$|f(x_i)|$$ line. If residuals remain very small, the computed complex roots are numerically reliable.
Practical Input Guidance for Numerical Accuracy
Keep coefficient scales reasonably balanced when possible. Extremely uneven magnitudes can increase floating-point sensitivity in iterative polynomial solvers. When needed, scale coefficients first, compare root patterns, and use residual output to validate confidence.
Keep exploring Algebra, Math Calculators, and Calculators.
Questions About the Depressed Quartic Solver
Quick answers about quartic roots, complex solutions, and numerical residual accuracy.
A depressed quartic is a fourth-degree polynomial with no cubic term, commonly written as $a x^4 + c x^2 + d x + e = 0$ with $a \neq 0$.
It normalizes the equation to $x^4 + p x^2 + q x + r = 0$, computes roots numerically, and reports residual checks $|f(x_i)|$ to verify solution quality.
Quartic polynomials do not always have four real roots. Complex values are expected and usually appear as conjugate pairs $u \pm vi$.
Max residual is the largest value of $|f(x_i)|$ among computed roots. Smaller values indicate stronger numerical agreement with the original equation.
No. This tool is for depressed quartic form only. Equations with an $x^3$ term should be transformed first or solved with a general quartic workflow.
Ensure $a \neq 0$, avoid extreme coefficient scaling when possible, and use residual checks to validate computed roots.