The formula to solve n degree equation

[NaxAlpha]

Hi

I want to solve 3 or 4 or up to n degree equation

I tried many time to solve only 3 degree equation but cannot solve

ax³+bx²+cx+d=0

x=?

a₀xⁿ+a₁x^n-1+a₂x^n-2+....+a_n-1x+a_n=0

what is general formula to calculate value of x?

I'm not talking about calculus which can give value of x by Newton's method I want algebraic solution!

[D H]

There is no general purpose formula. There are order-specific techniques for linear, quadratic, cubic, and quartic equations, and that's it. The wikipedia pages on cubic and quartic equations show how to solve these equations. Warning: Cubic equations are a mess, quartics, even messier.

 

That there is no general purpose formula for finding the roots of a fifth degree polynomial (or higher) is the result of Abel's impossibility theorem, one of the more celebrated theorems in mathematics. An alternative way to show that no formula is via Galois theory, a very important subject of abstract mathematics.

[Amaton]

Like the quadratic formula for degree-2, there exists a formula for the roots of a cubic. However, it's pretty nasty. There's no easy way of simplifying it into a single expression like using the [math]\pm[/math] in the quadratic formula.

 

There are some advanced methods of calculating roots directly, but I'm only familiar with more elementary approaches. These can get the job done for most polynomial equations: factoring, rational root theorem, synthetic division.

Edited January 12, 2013 by Amaton

[x(x-y)]

Indeed there is a "cubic formula", however it is rather more difficult to derive than the quadratic formula and I wouldn't recommend memorising it.