# Using Newton's Method without a calculator

#### SyNtHeSiS

How would you use Newton's Method without a calculator, if you are for example asked to to find a solution correct to 3 decimal places? I can only do it as a fraction, and my exam coming up soon and we not allowed to use calculators.

#### pickslides

MHF Helper
post a question, I will show you how.

#### SyNtHeSiS

Use Newton's method with the specified initial approximation x1 to find x3, the third approximation to the root of the given equation. (Give your answer to four decimal places):

5. x^3 + 2x - 4 = 0, x1 = 1

#### pickslides

MHF Helper
$$\displaystyle f(x)=x^3 + 2x - 4 \implies f'(x)=3x^2 + 2$$

Newton's method $$\displaystyle x_{n+1} = x_n-\frac{f(x_n)}{f'(x_n)}$$

For $$\displaystyle x_1 = 1$$

$$\displaystyle x_{2} = x_1-\frac{f(x_1)}{f'(x_1)}=1-\frac{(1)^3 + 2(1) - 4)}{(3(1)^2 + 2}= 1-\frac{1+2-4}{3+2}= 1-\frac{-1}{5} = \frac{6}{5}$$

Your turn. Find $$\displaystyle x_2$$

#### SyNtHeSiS

Yeah I got that, then I found x3 to be 932/790, but not sure of an efficient way of showing it to 4 decimal places without a calculator, besides using long division (which may indeed be very long lol)

#### HallsofIvy

MHF Helper
The only way to show $$\displaystyle \frac{932}{789}$$ "to four decimal places" is to divide.