Question
Download Solution PDFThe 2nd approximation to a root of the equation x2 - x - 1 = 0 in the interval (1, 2) by Bisection method will be:
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Bisection method:
Used to find the root for a function. Root of a function f(x) = a such that f(a)= 0
Property: if a function f(x) is continuous on the interval [a…b] and sign of f(a) ≠ sign of f(b). There is a value c belongs to [a…b] such that f(c) = 0, means c is a root in between [a….b]
Note:
Bisection method cut the interval into 2 halves and check which half contains a root of the equation.
1) Suppose interval [a…b] .
2) Cut interval in the middle to find m : \(m =\frac{{a+b}}{{2}}\)
3) sign of f(m) not matches with f(a) proceed the search in the new interval.
Calculation:
Given:
f(x) = x2 - x - 1 = 0 , a = 1 , b = 2
f(1) = 1 - 1 -1 = -1 < 0 , f(2) = 22 - 2 - 1 = 1 > 0 , Hence root lies between 1 and 2
By Bi-section method, \(x_0 = \frac{{a+b}}{{2}} = \frac{3}{2}\)
\(f(x_0) =\frac{9}{4}-\frac{3}{2}-1=-\frac{1}{4}<0\)
Which is positive. Hence, the root lies between 1.5 and 2
By Bi-section method, \(x_1 = \frac{{2+1.5}}{{2}} = 1.75\)
The 2nd approximation to a root of the equation x2 - x - 1 = 0 in the interval (1, 2) by Bisection method is 1.75
Last updated on Jul 16, 2025
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