# How do you use Newton's Method to approximate the root of the equation #x^4-2x^3+5x^2-6=0# on the interval #[1,2]# ?

##### 1 Answer

The answer is

Recall that Newton's Method uses the formula:

#x_(n+1)=x_n−(f(x_n))/(f'(x_n))#

The equation is already a function, so:

#f(x)=x^4-2x^3+5x^2-6#

And we need the derivative:

#f'(x)=4x^3-6x^2+10x#

The easiest way to iterate is to program your calculator. Enter

#A−(Y_1(A))/(Y_2(A))->A#

You can go to my website for specific instructions for the TI-83 or the Casio fx-9750 .

Finally, you need a starting value,

#1->A#

Then execute the program until you get the desired accuracy:

#1.2625#

#1.218807774#

#1.217563128#

#1.217562155#

#1.217562155#

We get 3 digits of accuracy after 2 iterations, 6 after 3 iterations, and 10 after 4 iterations. So the answer converges very quickly for this root.