Question

The root of equation x³ – 4x – 9 = 0 using the bisection method is
1. 1.6875
2. 2.6875
3. 3.6875
4. 4.6875

Answer 1

Correct Answer - Option 2 : 2.6875

Concept:

Bisection method:

• Used to find the root for a function.
• Property: Let a function f(x) is continuous on the interval [a,b] and sign of f(a) ≠  sign of f(b) or Such that f(a)×f(b)< 1. Then, there is a value c belongs to [a,b] such that f(c) = 0, means c is a root in between [a,b]

Calculation:

Given:
f(x) = x³ - 4x - 9 = 0
First of we have to assume an internal \(\left[ {a,b} \right]\), such that f(a)×f(b) < 0

So, we assume b = 3 and a = 1

f(b) = 3³ - 4× 3 - 9 = 6,

f(a) = 1 - 4 - 9 = - 12 

f(a)×f(b) = 6 × -12 = - 48 < 0,  So our assumption for the interval was correct.

C₁ = \(\frac{{a + b}}{2}\) = \(\frac{{3 + 1}}{2}\) = 2 ---------( 1^st iteration )

f(C₁) = 2³ - 8 - 9 = - 9

C₂ = \(\frac{{a + b}}{2}\) = \(\frac{{2 + 3}}{2}\) = 2.5 -------( 2^nd iteration)

f(C₂) = 2.5³ - 4×2.5 - 9 = - 3.375

C₃ = \(\frac{{a + b}}{2}\) = \(\frac{{2.5 + 3}}{2}\) = 2.75 --------- ( 3^rd iteration )

f(C₃) = 0.79

C₄ = \(\frac{{a + b}}{2}\) = \(\frac{{2.5 + 2.75}}{2}\) = 2.625--------( 4^th iteration )

f(C₄) = 2.67³ - 4×2.67 - 9 ≈ 0

The root of given equation is 2.625.

Note: These questions are very lengthy when we solve them by regular method, but in the exam, to save time we can use the elimination technique for finding the root of the answer, to solve the problem.