Quintic Equation — Definition, Formula & Examples

A quintic equation is a polynomial equation of degree 5, meaning the highest power of the variable is 5. By the Fundamental Theorem of Algebra, every quintic equation has exactly five roots (counted with multiplicity) in the complex numbers.

A quintic equation is an equation of the form $a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0$ , where $a_5 \neq 0$ and the coefficients $a_0, a_1, \ldots, a_5$ belong to a given field (typically $\mathbb{R}$ or $\mathbb{C}$ ). The Abel–Ruffini theorem proves that no general algebraic formula in radicals exists for solving arbitrary quintic equations, unlike quadratics, cubics, and quartics.

Key Formula

a_5x^5 + a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0 = 0

Where:

$a_5$ = Leading coefficient (must be nonzero)
$a_4, a_3, a_2, a_1, a_0$ = Remaining coefficients, which may be any real or complex numbers
$x$ = The unknown variable

How It Works

You can solve specific quintic equations by factoring, but there is no universal formula using only addition, subtraction, multiplication, division, and radicals that works for every quintic. When a quintic factors nicely — for instance, when you can pull out a common factor of

x

or identify a rational root via the Rational Root Theorem — you reduce it to lower-degree polynomials you already know how to solve. For quintics that do not factor over the rationals, numerical methods (like Newton's method) or elliptic functions are used to find roots.

Worked Example

Problem: Find all roots of the quintic equation

x^5 - 5x^3 + 4x = 0

Factor out x: Every term contains

x

, so factor it out.

x(x^4 - 5x^2 + 4) = 0

Substitute to reduce degree: Let

u = x^2

. The quartic factor becomes a quadratic in

u

u^2 - 5u + 4 = 0 \implies (u - 1)(u - 4) = 0

Solve for x: From

u = 1

x^2 = 1

, so

x = \pm 1

. From

u = 4

x^2 = 4

, so

x = \pm 2

. Combined with the factored

x = 0

, we have five roots.

x \in \{-2,\, -1,\, 0,\, 1,\, 2\}

Answer: The five roots are

x = -2, -1, 0, 1, 2

Why It Matters

The quintic equation is historically significant because proving it has no general radical solution (Abel–Ruffini theorem, 1824) launched the field of Galois theory, a cornerstone of abstract algebra. In applied mathematics and engineering, higher-degree polynomials including quintics arise in control theory and curve fitting, where numerical root-finding is standard practice.

Common Mistakes

Mistake: Assuming a formula like the quadratic formula exists for quintic equations.

Correction: The Abel–Ruffini theorem proves no such general radical formula exists for degree 5 or higher. You must rely on factoring, the Rational Root Theorem, or numerical methods.