Question

Show that the equation 3x^2 - x^3 + 3 = 0 can be rearranged to give:

x = 3 + 3
----
x^2

x^3 =

Divide by ? = x = 3 + 3
----
x^2

Using Xn+1 = 3 + 3 with x0 = 3.2
----
x^2n


find the values of x1 x2 and x3?

What to the values of x1 x2 and x3 represent?

The represent estimates of a _____ to the equation?

Answer 1

Just re-arrange it ....
3x^2 - x^3 + 3 = 0
-x^3 = -3x^2 - 3
x^3 = 3x^2 + 3
divide both sides by x^2

x = 3 + 3/x^2
let x = 3.2
RS = 3+ 3/10.24 = 3.29296875
sub that into the RS
RS = 3 + 3/(3.2929...)^2 = 3.276659...
new RS = 3.279420685
new RS = 3.278950402
new RS = 3.279030424
new RS = 3.279016806
new RS = 3.279019123
new RS = 3.279018729
new RS = 3.279018796
ok, once more
new RS = 3.279018784 , good enough?, we could get closer!

Wolfram shows that this is the only real solution
http://www.wolframalpha.com/input/?i=y+%3D+3x%5E2+-+x%5E3+%2B+3

If you need the two complex roots, use synthetic division to reduce your cubic to a quadratic and use the formula

answer confirmed by
http://www.wolframalpha.com/input/?i=solve+3x%5E2+-+x%5E3+%2B+3+%3D+0

click on "exact forms" and then even on "more digits" to show how close we got