Solving Equations: How to Find the Solution for (4/3)x^3 + x^2 - 2x + 8/3 = 0

[kd001]

Homework Statement

This isn't really homework. I've just been practicing some equations and I'm stuck at the one below.

Homework Equations

(4/3)x^3 + x^2 - 2x + 8/3 = 0

The Attempt at a Solution

The quadratic formula does not apply to this so how do I solve it for x?

Thanks

 

[slider142]

The first thing I would do is multiply both sides by 3/4 to get rid of the ugly leading term. Then try using the rational roots theorem. In short, if there is a rational root for your polynomial, then it is given by a factor of the coefficient of x⁰ over a factor of the coefficient of x³. This will hopefully give you a very short list to try. If none of those work, then the polynomial has no rational roots and you have to use more advanced methods. If it does work, you can use long division to get the rest of the roots, if any.

 

[kd001]

Thanks for the quick reply. I'll try that although its not something I'm familiar with.

 

[slider142]

Thanks for the quick reply. I'll try that although its not something I'm familiar with.

For cubics, there is always Cardano's formula, but it is rather messy in that it can give complicated complex expressions for simple real roots. The rational root method is often faster and less time consuming.
Otherwise, as you are posting in the calculus forum, you can use Newton-Raphson to approximate a root if there are no rational roots.