# Numerical method for solving Irrational Equation

How to solve the following equation:

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

I think it is hard. Thanks.

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You can use Newton's method –  Patrick Li Oct 12 '12 at 12:05
$x=0$ ${ }$ ${ }$ ${ }$ –  draks ... Oct 12 '12 at 12:17
@draks $x=0$ is only one of the solutions. There are more. –  Patrick Li Oct 12 '12 at 12:21
I checked that this has 3 roots. $x=0$ is easy to see, how about 2 other ones? –  newday Oct 12 '12 at 12:48

There are only three roots, in $\{-u,0,u\}$, with $u$ pretty close to $1$. By convexity, the Newton method with starting point $x_0=\pm 1$ has quadratic convergence.