# Impossible to solve for x. What to do now?

1. Feb 7, 2010

### Juwane

1. The problem statement, all variables and given/known data

Solve this equation:

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

3. 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?

2. Feb 7, 2010

3. Feb 7, 2010

### vela

Staff Emeritus
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.

4. Feb 7, 2010

### Juwane

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