Question

Roots of 4x^3 + 4x^2 + x =0

Answer 1

Step-by-step explanation:

We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation).

The function is F(x) = 4x3 - 4x2 - x - 1

At x= 1.00 F(x) is equal to -2.00

At x= 2.00 F(x) is equal to 13.00

Intuitively we feel, and justly so, that since F(x) is negative on one side of the interval, and positive on the other side then, somewhere inside this interval, F(x) is zero

Procedure :

(1) Find a point "Left" where F(Left) < 0

(2) Find a point 'Right' where F(Right) > 0

(3) Compute 'Middle' the middle point of the interval [Left,Right]

(4) Calculate Value = F(Middle)

(5) If Value is close enough to zero goto Step (7)

Else :

If Value < 0 then : Left <- Middle

If Value > 0 then : Right <- Middle

(6) Loop back to Step (3)

(7) Done!! The approximation found is Middle

Follow Middle movements to understand how it works :

Left Value(Left) Right Value(Right)

1.000000000 -2.000000000 2.000000000 13.000000000

0.000000000 -1.000000000 2.000000000 13.000000000

1.000000000 -2.000000000 2.000000000 13.000000000

1.000000000 -2.000000000 1.500000000 2.000000000

1.250000000 -0.687500000 1.500000000 2.000000000

1.250000000 -0.687500000 1.375000000 0.460937500

1.312500000 -0.159179688 1.375000000 0.460937500

1.312500000 -0.159179688 1.343750000 0.139038086

1.328125000 -0.012985229 1.343750000 0.139038086

1.328125000 -0.012985229 1.335937500 0.062292099

1.328125000 -0.012985229 1.332031250 0.024470568

1.328125000 -0.012985229 1.330078125 0.005697042

1.329101562 -0.003655490 1.330078125 0.005697042

1.329101562 -0.003655490 1.329589844 0.001017926

1.329345703 -0.001319494 1.329589844 0.001017926

1.329467773 -0.000150962 1.329589844 0.001017926

1.329467773 -0.000150962 1.329528809 0.000433437

1.329467773 -0.000150962 1.329498291 0.000141226

1.329483032 -0.000004871 1.329498291 0.000141226

1.329483032 -0.000004871 1.329490662 0.000068177

1.329483032 -0.000004871 1.329486847 0.000031653

1.329483032 -0.000004871 1.329484940 0.000013391

Next Middle will get us close enough to zero:

F( 1.329483509 ) is -0.000000305

The desired approximation of the solution is:

x ≓ 1.329483509