The approximate solution of the provided equation after three iterations of successive approximations is when x is about is -37/16.
What is iteration method?
The Iteration method is the process which takes the initial value to produce the series of successive approximations of the solution of the equation.
The equation given in the problem is,
[tex]\dfrac{2}{x+4}= 3^x + 1[/tex]
Solve it further,
[tex]-3^x -1+\dfrac{2}{x+4}= 0[/tex]
Let the above function if f(x),
[tex]f(x)=-3^x - 1+\dfrac{2}{x+4}[/tex]
The differentiations of the above function is,
[tex]f'(x)=-3^x \ln3-\dfrac{2}{(x+4)^2}[/tex]
Here, for this function
x 0, -1, -2, -3
x -1.5, -0.667, -0.111, 0.963
Here, the function,
[tex]f(-3)=0.963 > 0 \;\;\;\text{and}\\f(-2)=-0.111 < 0[/tex]
Hence, the roots lies between -3 and -2.
[tex]x_o=\dfrac{-3+-2}{2}\\x_o=-2.5[/tex]
This value is more near to the value -37/16 from the given option.
Thus, the approximate solution of provided equation after three iterations of successive approximations is when x is about is -37/16.
Learn more about the iteration method here;
https://brainly.com/question/24480057
