Formulas to solve Polynomial Equations.

The general form of the nth degree equation is:  a0xn + a1xn-1 + a2xn-2 + ... + an-1x +  an = 0

The nth degree equations have always n roots.  In particular cases, some or all of this n roots could be equal to one another.

If the coefficients ai are real numbers, then the roots could be real or complex  numbers.  (Any combination, with the following restriction: if one of the roots is complex, then its conjugate is also a root.  This implies that complex roots comes in pairs and that odd degree equations have at least one real root.)

First degree equations:
ax + b = 0

One root:

Second degree equations (or Quadratics):

ax2 + bx + c = 0



Third degree equations (or Cubics):
ax3 + bx2 + cx + d = 0

Three roots:
Press here to see them (the formulas are really big).

Fourth degree equations (or Quartics):
ax4 + bx3 + cx2 + dx + e = 0

Four roots:
Press here to see them (the formulas are even bigger).

Equations of degree higher than four:
The roots of equations of degree higher than four can't, in general, be expressed using only the operations of addition, subtraction, multiplication, division and extraction of nth roots [Ruffini, Abel, Galois].  However, these roots can be found with numerical algorithms.

First post in: 2002
Last update
: 2006-02-04