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.
-$$3 x ^ { 2 } + 2 x - 1 = 0 ; x _ { 0 } = 1 ; \text { Find the right-hand solution. }$$
A) 0.33
B) 0.85
C) 0.35
D) 0.50

Question

Quizplus · Accepted Answer

Newton's method: $$x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$$$$f(x)=3x^2+2x-1$$$$f'(x)=6x+2$$$$x_0=1$$$$x_1=x_0-\frac{f(x_0)}{f'(x_0)}=1-\frac{3(1)^2+2(1)-1}{6(1)+2}=1-\frac{4}{8}=1-\frac{1}{2}=\frac{1}{2}$$$$x_2=x_1-\frac{f(x_1)}{f'(x_1)}=\frac{1}{2}-\frac{3\left(\frac{1}{2}\right)^2+2\left(\frac{1}{2}\right)-1}{6\left(\frac{1}{2}\right)+2}=\frac{1}{2}-\frac{\frac{3}{4}+1-1}{\frac{6}{2}+2}=\frac{1}{2}-\frac{\frac{3}{4}}{\frac{10}{2}}=\frac{1}{2}-\frac{3}{20}=\frac{10}{20}-\frac{3}{20}=\frac{7}{20}=0.35$$

Use Newton's Method to Estimate the Requested Solution of the Equation