Use Newton's method to estimate the requested solution of the equation. Start with given value of X0 and then give x2 as the estimated solution. x3 + 5x +2 = 0; x0 = -1; Find the one real solution.

Step-by-step explanation:

Given the initial value of X0 = -1, we can determine the solution of the equation x³ + 5x +2 = 0 using the Newton's method. According to newton's approximation formula;

If

We will iterate using the formula;

Given f(x) = x³ + 5x +2

f(x0) = f(-1) = (-1)³ + 5(-1) +2

f(-1) = -1 -5 +2

f(-1) = -4

f'(x) = 3x²+5

f'(-1) = 3(-1)²+5

f'(-1) = 8

f(-0.5) = (-0.5)³ + 5(-0.5) +2

f(-0.5) = -0.125-2.5+2

f(-0.5) = -0.625

f'(-0.5) = 3(-0.5)²+5

f'(-0.5) = 3(0.25)+5

f'(-0.5) = 0.75+5

f'(-0.5) = 5.75

The estimated solution is -0.3913 (to 4dp)