How to solve a quartic equation with $x^4$ and $x$?

Question

I am solving some heat transfer problems, and I came across this equation:

$$(4.536 \cdot 10^{-8})x^4+ 12 x - 4316 = 0$$

The solution is $x = 320$ (I have the solutions book).

I am using a HP50g calculator that gives this same value ($320$), but I need to know how to solve it manually.

Could you help me to understand a step by step of getting to this result? Can Casio scientific calculators solve it?

Answer 1

You can use Newton's approximation, which I believe is the one used by your calculator too.

Let $f(x)=Ax^4+12x-4316$, where $A=4.536\times10^{-8}$. To approximate $x$, after sufficient iterations, you can use the recursive function below: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$ Therefore you get: $$x_{n+1}=x_n-\frac{Ax_n^4+12x_n-4316}{4Ax_n^3+12}$$ After sufficient iterations, you reach $x=320.0203904...$

Answer 2

Since $A$ is so small, you can use a fixed-point iteration: $$ x_{n+1} = \frac{4316-Ax_n^4}{12} $$ Starting with $x_0=0$ gives $x_{20} \approx 320.0203$.