Impossible to solve for x. What to do now?

[Juwane]

Homework Statement

Solve this equation:

$$x^2 = 2 \sqrt{x^3 + 1}$$

The Attempt at a Solution

Squaring both sides:

$$( x^2 )^2 = ( 2 \sqrt{x^3 + 1} )^2$$

$$x^4 = 4( x^3 + 1 )$$

$$x^4 = 4x^3 + 4$$

$$x^4 - 4x^3 = 4$$

Now what? Is there is any technique in whole of mathematics with which we can find an apporiximate solution, if not the actual value?

 

[radou]

A bit of a long text, but I hope it will help: http://en.wikipedia.org/wiki/Quartic_function#Solving_a_quartic_equation

 

[vela]

You could use Newton-Raphson to find the roots. If you have an approximation $x_n$ for the root, you can get a new approximation by calculating

$$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$

When it works, it converges on the root quickly. You can get initial guesses by plotting the function.

 

[Juwane]

Is Newton-Raphson method the best method available for approximation?