How to solve the cubic equation $-1.6x^3-2.1x^2-2.9x-0.3=0$

Question

Is there any method similar to the quadratic formula to solve the equation;

{$-1.6x^3-2.1x^2-2.9x-0.3=0$}

If not, how would I go about finding the solutions for cubic equations like these?

Answer 1

There are two good numerical methods, that give the answer as accurately as you like.

The first is the ' bisection method'. You start with a broad interval that contains a aolution, then narrow it down as finely as you like.

The second is 'Newton's Method'. You need calculus to set it up, but then it is usually quicker than Bisection. In this case, the key formula is

$$x_{n+1}=x_n-\frac{1.6x_n^3+2.1x_n^2+2.9x_n+0.3}{4.8 x_n^2+4.2x_n+2.9}$$

Start with any value for $x_1$. Calculate $x_2$ from $x_1$, then $x_3$ from $x_2$ and so on. If you aren't unlucky, it will home in on a solution in three or four steps.