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Once a root is found, it’s usually possible to improve the precision of the root.

So basically this is getting a more precise/accurate number for the root

The common method for this is called Interval Bisection , this is when the interval is cut in half after each iteration (process with an input and an output which is repeated)

practice question

(i) Verify that a root of the equation 2^x - 11 - 5x = 0 lies in the interval 5 < x < 5.8

step 1: substitute 5 and 5.8 as x and find the value

f(5) = -4 < 0 f(5.8) = 15.7 > 0

as you can see one is positive and one is negative meaning the root is in the interval

(ii) Use interval bisection to find an interval of width 0.1 that contains the root

step 1: find the number (in this case to 1.d.p) in the middle of the 2 intervals, and substitute it as x

f(5.4) = 4.2 > 0

step 2: keep substituting numbers until you get 2 close to 0, 1 positive and 1 negative

f(5.3) = 1.9 > 0

f(5.2) = -0.2 < 0

step 3: rewrite the interval with the new more precise numbers

answer: 5.2 < x < 5.3