Answer to Question #200908 in Quantitative Methods for George

Question

Find the negative root of the equation x^3 - 2x + 5 =0

Answer 1

Find a root of an equation f(x)=x3-2x+5 using Bisection method

Solution:

Here x³-2x+5=0

Let f(x)=x³-2x+5

Here

x           0            -1           -2           -3           -4
f(x)        5             6           1           -16           -51

1st iteration :

Here f(-3)=-16<0 and f(-2)=1>0

∴ Now, Root lies between -3 and -2

x₀=-3+(-2)2=-2.5

f(x₀)=f(-2.5)=(-2.5)3-2⋅(-2.5)+5=-5.625<0

2nd iteration :

Here f(-2.5)=-5.625<0 and f(-2)=1>0

∴ Now, Root lies between -2.5 and -2

x₁=-2.5+(-2)2=-2.25

f(x₁)=f(-2.25)=(-2.25)3-2⋅(-2.25)+5=-1.8906<0

3rd iteration :

Here f(-2.25)=-1.8906<0 and f(-2)=1>0

∴ Now, Root lies between -2.25 and -2

x₂=-2.25+(-2)2=-2.125

f(x₂)=f(-2.125)=(-2.125)3-2⋅(-2.125)+5=-0.3457<0

4th iteration :

Here f(-2.125)=-0.3457<0 and f(-2)=1>0

∴ Now, Root lies between -2.125 and -2

x₃=-2.125+(-2)2=-2.0625

f(x₃)=f(-2.0625)=(-2.0625)3-2⋅(-2.0625)+5=0.3513>0

5th iteration :

Here f(-2.125)=-0.3457<0 and f(-2.0625)=0.3513>0

∴ Now, Root lies between -2.125 and -2.0625

x₄=-2.125+(-2.0625)2=-2.0938

f(x₄)=f(-2.0938)=(-2.0938)3-2⋅(-2.0938)+5=0.0089>0

6th iteration :

Here f(-2.125)=-0.3457<0 and f(-2.0938)=0.0089>0

∴ Now, Root lies between -2.125 and -2.0938

x₅=-2.125+(-2.0938)2=-2.1094

f(x₅)=f(-2.1094)=(-2.1094)3-2⋅(-2.1094)+5=-0.1668<0

7th iteration :

Here f(-2.1094)=-0.1668<0 and f(-2.0938)=0.0089>0

∴ Now, Root lies between -2.1094 and -2.0938

x₆=-2.1094+(-2.0938)2=-2.1016

f(x₆)=f(-2.1016)=(-2.1016)3-2⋅(-2.1016)+5=-0.0786<0

8th iteration :

Here f(-2.1016)=-0.0786<0 and f(-2.0938)=0.0089>0

∴ Now, Root lies between -2.1016 and -2.0938

x₇=-2.1016+(-2.0938)2=-2.0977

f(x₇)=f(-2.0977)=(-2.0977)3-2⋅(-2.0977)+5=-0.0347<0

9th iteration :

Here f(-2.0977)=-0.0347<0 and f(-2.0938)=0.0089>0

∴ Now, Root lies between -2.0977 and -2.0938

x₈=-2.0977+(-2.0938)2=-2.0957

f(x₈)=f(-2.0957)=(-2.0957)3-2⋅(-2.0957)+5=-0.0129<0

10th iteration :

Here f(-2.0957)=-0.0129<0 and f(-2.0938)=0.0089>0

∴ Now, Root lies between -2.0957 and -2.0938

x₉=-2.0957+(-2.0938)2=-2.0947

the negative root of the equation =-2.0947