Question

Use the Bisection Algorithm Method to find the root of the given function to an interval of length less than or equal to 0.1.  Answer should be up to one decimal place only.

f(x) = x2 - 2 on [0,3]

Answer 1

f(0)=-2, f(3)=9-2=7. We want to find the value of x such that f(x)=0.

Since 0 lies between -2 and 7, we divide the interval in half and calculate f(1.5)=1.5²-2=2.25-2=0.25.

This result is greater than zero so we now halve the interval between 0 and 1.5, because f(0)<0 and f(1.5)>0, so f(x)=0 somewhere between x=0 and 1.5. Bisect this interval and x=0.75.

f(0.75)=-1.4375. Therefore the root lies between x=0.75 and 1.5. Halfway between these is their average=1.125. f(1.125)=-0.734375. The root lies between 1.125 and 1.5, which average to 1.3125.

f(1.3125)=-0.27734375. The root lies between x=1.3125 and 1.5, which average to 1.40625. f(1.40625)=-0.0224609375. But this is less than the error of ±0.1, so x=1.4 is an approximate root of the function.