Q5. Find the real root of the equation f(x) = 4x³+3x²-4x-5 using bisection method. Perform three reiterations.

Solution:

f(x) = 4x³+3x²-4x-5

Consider The following table of value

x	0	1	2
f(x)	-5	-2	31

take a1=1, b1=2, Since f(a₁)f(b₁) < 0

we give the four iteration in the following tabular form.

N	an	bn	αn	f(αn)
1	1	2	1.5	10.75
2	1	1.5	1.25	2.58
3	1.25	1.5	1.375	5.57031
4	1.25	1.375	1.3125	3.9619

after three iteration , we have α₄ = 1.3125 approximating

the root α with an error

|α – α₃| <= |1.375 – 1.325| = 0.50 and since 1.3125 < α.

|α – α₄| / |α| < |b₄ – a₄| / |a₄| < 0.5/10 = ½ x 10^-1 = ½ x 10^1-2

That is, the approximation is correct to at least to at least 2sinnificant digits.

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