Answer to Question #200888 in Quantitative Methods for George

Question

Find the real roots of the equation x^3 + x^2 - 100= 0 by the iteration method

Answer 1

$4^3+4^2-100<0$

$5^3+5^2-100>0$

So, the root is between 4 and 5.

$x^2(x+1)=100$

$x=\frac{10}{\sqrt{x+1}}$

Let the initial approximation be $x_0=4.5$

$x_1=10/\sqrt{4.5+1}=4.26$

$x_2=10/\sqrt{4.26+1}=4.35$

$x_3=10/\sqrt{4.35+1}=4.32$

$x_4=10/\sqrt{4.32+1}=4.33558$

$x_5=10/\sqrt{4.33558+1}=4.32921$

$x_6=10/\sqrt{4.32921+1}=4.33180$

$x_7=10/\sqrt{4.33180+1}=4.33117$

Since the difference between x₆ and x₇ are very small, so the root is 4.33117