## Equations of 4-th Degree

The program determines the real solutions of an equation of the 4th or of a lower degree.

**a·x**^{4} + b·x^{3} + c·x^{2} + d·x + e = 0

### Example:

To determine the solutions of the equation x^{4} + 2x^{3} - 8x^{2} -18x - 9 = 0 ,
enter the coefficients a to e as follows:

and receive the set of solutions:

x^4 + 2x^3 - 8x^2 - 18x - 9 = 0
L = {-1; -3; 3}

The formula for quadratic equations is well known. The formula for cubic equations
was derived by *Scipione del Ferro* in 1530, but published after his death by his adept
*Ceralamo Cardano* . The extension to equations of 4-th degree Cardano himself attributed to his adept
*Lodovico Ferrari * .

### Equations of 5th or higher degree

We have to thank the Norwegian mathematician *Niels Henrik Abel* for the proof that no formula can exist for equations
with higher degree than 4. The only solutions are approximations like we do in curve discussions.

If we know one solution of an equation, we sometimes can find the others by polynomial division.

For example x^{5} - 12x^{3} - 2x^{2} + 27x + 18 = 0
has the solution x_{1} = 2 .

Thus we know that the left site of the equation can be divided by (x - 2) without a remainder.

(- 18x^{5} + 45x^{4} - 2x^{3} - 12x^{2} + 1 ) : (x - 2 ) = x^{4} + 2x^{3}- 8x^{2} - 18x - 9

The equation x^{4} + 2x^{3}- 8x^{2} - 18x - 9 = 0 allows us to find the missing solutions.

### See also:

Wikipedia:Cubic function
Wikipedia: Scipione del Ferro |

Gerolamo Cardano |

Ludovico Ferrari