Use bisection method to find the root of $2x(1-x^{2}+x)\ln(x)=x^{2}-1$ on [0,1]

Question

I have to use the bisection method to find the root of $2x(1-x^{2}+x)\ln(x)=x^{2}-1$ on $[0,1]$.

However I simplify I get,

$$x^{2x(1-x^{2}+x)}-e^{x^{2}-1}=0$$

Which is undefined at $0$. How would I go about approaching this problem?

Answer 1

I think the interval may be wrong as $\ln (0)$ is undefined from the start, but regardless the solution is $x=1$

Answer 2

We are being asked to use the Bisection Method to find the root of:

$$2x(1-x^{2}+x)\ln(x)=x^{2}-1, x\in [0,1]$$

If we plot these functions, we see two points of intersection:

Performing the Bisection Method using $20$ digits of accuracy using $a = 0.05, b = 0.35$ converges in $62$ steps to the root:

$$x = 0.32796778533181880526$$

Repeating this for the next root, performing the Bisection Method using $20$ digits of accuracy using $a = 0.8, b = 1.0$ converges in $31$ steps to the root:

$$x = 0.99999999990686774254$$