How to Find Polynomial Roots: Linear, Quadratic, Cubic, and Quartic

What Are Polynomial Roots?

The roots (or zeros) of a polynomial are the values of x where the polynomial equals zero. A polynomial of degree n has exactly n roots when counted with multiplicity in the complex number system — a fact guaranteed by the Fundamental Theorem of Algebra.

For a degree-2 polynomial ax² + bx + c = 0, there are 2 roots. Some may be real and distinct, some may repeat, and some may be complex numbers. Understanding how to find each type is fundamental to algebra, engineering, and physics.

Linear Equations (Degree 1)

For ax + b = 0, the root is immediate:

x = −b / a

Example: 2x + 6 = 0 → x = −6/2 = −3.

Quadratic Equations (Degree 2)

For ax² + bx + c = 0, use the quadratic formula:

x = (−b ± √(b² − 4ac)) / 2a

The expression under the square root — b² − 4ac — is called the discriminant D.

• D > 0: two distinct real roots
• D = 0: one repeated real root
• D < 0: two complex conjugate roots (a ± bi)

Worked Example

Polynomial: x² − 5x + 6 = 0 (coefficients a=1, b=−5, c=6)

Discriminant: D = (−5)² − 4(1)(6) = 25 − 24 = 1

Roots:

x = (−(−5) ± √1) / (2×1)
  = (5 ± 1) / 2

x₁ = (5 + 1) / 2 = 3
x₂ = (5 − 1) / 2 = 2

Result: Roots are x = 2 and x = 3.

Verification: (x − 2)(x − 3) = x² − 5x + 6. ✓

Numerical Stability in the Quadratic Formula

The standard formula x = (−b + √D)/(2a) can lose precision when b is very large relative to √D, because the subtraction cancels significant digits. A numerically stable form computes:

x₁ = (−b − sign(b)·√D) / 2a
x₂ = c / (a · x₁)      [Vieta's formula for the second root]

This avoids the catastrophic cancellation of the standard formula for large |b|.

Cubic Equations (Degree 3)

For ax³ + bx² + cx + d = 0, Cardano’s formula (1545) provides an algebraic solution. The algorithm first depresses the cubic (removes the x² term via a substitution) to get t³ + pt + q = 0, then solves using:

D = −4p³ − 27q²

• D > 0: three distinct real roots (casus irreducibilis — use trigonometric form)
• D = 0: a repeated root
• D < 0: one real root and one complex conjugate pair

The trigonometric form for the three-real-root case avoids the complex intermediate values that Cardano’s formula produces even when all roots are real.

Quartic Equations (Degree 4)

Ferrari’s method (1540) reduces a quartic to a resolvent cubic, then solves via the cubic formula. The quartic is factored into two quadratics whose coefficients come from the cubic’s roots. Each quadratic is then solved with the quadratic formula.

Complex Roots

When the discriminant of a quadratic is negative, the roots are complex conjugate pairs: p ± qi where p = −b/(2a) and q = √|D|/(2a). Complex roots always appear in conjugate pairs for polynomials with real coefficients — if a + bi is a root, then a − bi is also a root.

Complex roots do not correspond to x-intercepts on the real number line, but they are valid solutions in the complex number system and appear frequently in control theory, electrical engineering, and quantum mechanics.

Choosing the Right Method

For degree 5 and above, the Abel–Ruffini theorem (1824) proves that no general algebraic formula exists. Numerical methods (Newton’s method, Durand-Kerner) are used instead.