Quartic Equation Solver

About Quartic Equation Solver

The Quartic Equation Solver finds all four roots of any quartic (fourth-degree polynomial) equation in the form ax⁴ + bx³ + cx² + dx + e = 0. Enter the five coefficients and get instant results with a step-by-step solution using Ferrari's method, discriminant analysis, factored form, Vieta's relations, and an interactive graph.

How to Use the Quartic Equation Solver

1. Enter coefficients: Type the values of a, b, c, d, and e for your quartic equation ax⁴ + bx³ + cx² + dx + e = 0. The leading coefficient a must not be zero.
2. Click "Solve Quartic Equation" to compute all four roots.
3. View the roots: Each root is displayed with a label showing whether it is real or complex. Real roots appear in green cards, complex roots in blue.
4. Study the step-by-step solution: Follow Ferrari's method from the depressed quartic through the resolvent cubic to the final quadratic factorization.
5. Explore the graph: See the quartic function plotted with real roots marked in green.

What Is a Quartic Equation?

A quartic equation is a polynomial equation of degree four:

\(ax^4 + bx^3 + cx^2 + dx + e = 0\)

where \(a \neq 0\). By the Fundamental Theorem of Algebra, every quartic equation has exactly four roots (counting multiplicity), which may be real or complex numbers. Unlike cubic equations that always have at least one real root, a quartic can have 0, 2, or 4 real roots.

Ferrari's Method

Discovered by Lodovico Ferrari in 1540 (and published by his teacher Cardano in 1545), this is the classical method for solving quartic equations. It works by:

1. Depressing the quartic: Substituting \(x = t - \frac{b}{4a}\) to eliminate the cubic term, yielding \(t^4 + pt^2 + qt + r = 0\)
2. Introducing an auxiliary variable: Adding \(mt^2 + m^2/4\) to both sides and choosing \(m\) so the right side becomes a perfect square
3. Solving the resolvent cubic: The condition for a perfect square leads to a cubic equation in \(m\)
4. Factoring into quadratics: With the right \(m\), the quartic factors as \((t^2 + st + u_1)(t^2 - st + u_2) = 0\)
5. Applying the quadratic formula twice to find all four roots

The Discriminant of a Quartic

The discriminant of a quartic equation is a polynomial expression in the coefficients that determines the nature of the roots:

• \(\Delta > 0\): Either all four roots are real, or all four are complex (two conjugate pairs)
• \(\Delta < 0\): Exactly two real roots and two complex conjugate roots
• \(\Delta = 0\): The equation has at least one repeated root

The quartic discriminant is significantly more complex than the cubic discriminant, involving terms up to degree 6 in the coefficients.

Vieta's Formulas for Quartic Equations

If \(x_1, x_2, x_3, x_4\) are the four roots of \(ax^4 + bx^3 + cx^2 + dx + e = 0\), then:

• \(x_1 + x_2 + x_3 + x_4 = -\frac{b}{a}\)
• \(\sum_{i
• \(\sum_{i
• \(x_1 x_2 x_3 x_4 = \frac{e}{a}\) (product of all roots)

Special Cases

• Biquadratic (\(b = d = 0\)): \(ax^4 + cx^2 + e = 0\) — substitute \(u = x^2\) and solve the resulting quadratic
• Depressed quartic (\(b = 0\)): \(x^4 + cx^2 + dx + e = 0\) — already in simplified form for Ferrari's method
• Difference of squares: \(x^4 - k^2 = (x^2 + k)(x^2 - k)\)
• Perfect fourth power: \((x - r)^4 = x^4 - 4rx^3 + 6r^2x^2 - 4r^3x + r^4\)

Quartic Equations vs. Higher Degrees

The quartic is the highest-degree polynomial equation that can be solved by radicals (using only addition, subtraction, multiplication, division, and root extraction). This was proven by Abel in 1824 and expanded by Galois — general quintic (degree 5) and higher equations have no closed-form radical solution.

Applications of Quartic Equations

• Optics: Ray tracing through curved surfaces (intersection of rays with tori)
• Engineering: Euler-Bernoulli beam deflection equations, vibration analysis
• Physics: Quartic potential in quantum mechanics, coupled oscillator systems
• Computer graphics: Ray-torus intersection, Bezier curve analysis
• Geometry: Finding intersection of conics (ellipses, parabolas, hyperbolas)
• Control theory: Stability analysis of fourth-order systems

FAQ

What is a quartic equation?

A quartic equation is a polynomial equation of degree 4, written as ax⁴ + bx³ + cx² + dx + e = 0, where a is not zero. Every quartic has exactly four roots (counting multiplicity), which may be real or complex.

How does Ferrari's method work?

Ferrari's method solves quartic equations by first converting to a depressed quartic (removing the cubic term), then introducing an auxiliary variable via a resolvent cubic equation. Solving this cubic yields a value that allows the quartic to be factored into two quadratic equations, each of which is then solved using the quadratic formula.

What does the discriminant of a quartic equation tell you?

The discriminant determines the nature of the roots. If positive, all roots are either all real or all complex. If negative, there are exactly two real roots and two complex conjugate roots. If zero, the equation has at least one repeated root.

Can all four roots of a quartic equation be complex?

Yes, unlike cubic equations, a quartic equation with real coefficients can have all four roots be complex. In this case, the roots form two pairs of complex conjugates.

What is Vieta's formulas for quartic equations?

Vieta's formulas relate the four roots to the coefficients. For ax⁴+bx³+cx²+dx+e=0 with roots r1, r2, r3, r4: the sum of roots equals -b/a, the sum of products of pairs equals c/a, the sum of products of triples equals -d/a, and the product of all roots equals e/a.

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