Answer
$x = 1.28$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/1accd6ff-16aa-4480-b7be-ee43949bc187/result_image/1554730447.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T012816Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=0898e70f6a14f1ee07aed55e86049b90f6f90748d67d4edef40a7779f3d66928)
Work Step by Step
$2sin~2x-x^3+1 = 0$
$2sin~2x = x^3-1$
We can graph the two equations $~~y = 2sin~2x~~$ and $~~y = x^3-1~~$ to find the points of intersection.
The blue graph is $~~y = x^3-1~~$
The red graph is $~~y = 2sin~2x~~$
On the interval $[0,2\pi]$, we can see that the point of intersection occurs when $~~x = 1.28~~$
The solution over the given interval is:
$x = 1.28$
![](https://gradesaver.s3.amazonaws.com/uploads/solution/1accd6ff-16aa-4480-b7be-ee43949bc187/steps_image/small_1554730447.jpg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAJVAXHCSURVZEX5QQ%2F20240727%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240727T012816Z&X-Amz-Expires=900&X-Amz-SignedHeaders=host&X-Amz-Signature=e9171233d38294e588ca82c64b0ceaebf8abbc402160d2b3dd5e54db6111e92b)