Solution:
Let,
F(x)=x^3-4x-9=0
Now,
Put x=0:- 0^(3)-4*0-9= -9 (-ve)
Put x=1:- 1^(3)-4*1-9= -13 (-ve)
Put x=2:- 2^(3)-4*2-9= -9 (-ve)
Put x=3:- 3^(3)-4*3-9= 6 (+v)
Therefore the roots lie between 2 and 3:
1st stage:-
Hence,
x0= 2+3/2= 5/2
x0=2.5
now,
f(x0)=2.5^(3)- 4*2.5 -9=0
f(x0)= -3.375
So, the roots lie between x0 and 2(x0 which is 2.5):
2nd stage:-
Hence,
x1= 2.5+3/2=
x1=2.75
now,
f(x1)=2.75^(3)-4*2.75-9
f(x1)= 0.796
So, the roots lie between x0 and x1(which is 2.5 and 2.75):
3rd stage:-
Hence,
x2= 2.5+2.75/2
x2= 2.625
now,
f(x2)= 2.625^(3) – 4*2.625 – 9=
f(x2)=-1.412
So, the roots lie between x1 and x2(which is 2.75 and 2.625):
4th stage:-
Hence,
x3= 2.75+2.625/2=2.6875
x3=2.6875
now,
f(x3)=2.6875^(3) – 4*2.6875 – 9=
f(x3)= -0.347
here, the roots lie between x1 and x3:
5th stage:-
Hence,
x4= 2.75+2.06875/2
x4= 2.718
now,
f(x4)= 2.718^(3) – 4*2.718 – 9=
f(x4)= 0.207
here, the roots lie between x1 and x3:
6th stage:-
Hence,
X5= 2.6875+2.718/2
X5= 2.702
now,
f(x5)= 2.702^(3) – 4*2.702 – 9=
f(x5)= -0.081
here, the roots lie between x4 and x5:
7th stage:-
Hence,
X6= 2.702+2.718/2
X6= 2.71
now,
f(x6)= 2.71^(3) – 4*2.71 – 9=
f(x6)= 0.062
here, the roots lie between x5 and x6:
8th stage:-
Hence,
X7= 2.702+2.71/2
x7= 2.706
now,
f(x7)= 2.706^(3) – 4*2.706 – 9=
f(x7)= -0.009
here, the roots lie between x5 and x7:
9th stage:-
Hence,
X8= 2.702+2.706/2
X8= 2.706
now,
f(x8)= 2.706^(3) – 4*2.706 – 9=
f(x8)= -0.009
Hence, the roots lie between x8= 2.706 upto three decimal number.