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When x=0, f(0)=x^2-2=-2 and f(3)=9-2=7. So there is a change of sign between x=0 and x=3, implying that the root lies in between.

The average or midway position is x=1.5 and f(1.5)=2.25-2=0.25. Therefore the root lies between 1.5 and 0.

The midway position is 0.75 and f(0.75)=0.5625-2=-1.4375. So the root lies between 0.75 and 1.5, i.e., 1.125 as the midway point. And f(1.125)=-0.734375. The midway point including the root is (1.125+1.5)/2=1.3125.

f(1.3125)=-0.27734375, meaning that the root lies between 1.3125 and 1.5, with a midpoint of 1.40625, and f(1.40625)<0, so the root lies in the interval [1.40625,1.5], the midway point being 1.453125, and f(1,453125)>0 so the root lies in the interval [1.40625,1.43125], the midway point of which is 1.41875. We can see that to one decimal place the root is 1.4, because both extremes of the limit give us 1.4.

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Hoping to help for my other post. Thanks again!