Use Newton's method with initial approximation x1 = −1 to find x2, the second approximation to the root of the equation x3 + x + 8 = 0. (Round your answer to four decimal places.) x2 =

The second approximation to the root of the equation is -1.5000.

Step-by-step explanation:

The Newton's method is a numerical method by approximation that help find roots of a equation of the form with the help of the equation itself and its first derivative. The Newton's formula is:

Where:

- i-th approximation, dimensionless.

- (i+1)-th approximation, dimensionless.

- Function evaluated at the i-th approximation, dimensionless.

- First derivative of the function evaluated at the i-th approximation, dimensionless.

The function and its first derivative are and , respectively. Now, the Newton's formula is expanded:

If , the value of is:

The second approximation to the root of the equation is -1.5000.

it would be 6xy and -16xy.