1. Find the root of  x2 - 10x + 23 = 0

a[]       1.0     -10.0     23.0

b[]       1.0     54.0     529.0

roots =    7.35     3.129

b[]        1.0     1858.0     279841.0

roots =    6.565     3.503

b[]        1.0     2892482.0     7.831E10

roots =   6.4218     3.585

b[]       1.0     8.2098E12     6.1326E21

roots =  6.414      3.585

6.414

3.585

Thus the absolute values of the roots are 6.414 and 3.585.

Since f(6.414) = 0 and f(3.585) = 0, the signs of the roots 6.414 and 3.585 are all positive.

 So one of the root of the give equation is 1.0

2. Find the root of  x + 3x2 - 4 = 0

a[]   1.0     3.0     -0.0     -4.0

b[]   1.0     9.0     24.0     16.0

roots =  3.0     1.633     0.816

b[]   1.0     33.0     288.0     256.0

roots =  2.39     1.718     0.971

b[]   1.0     513.0     66048.0     65536.0

roots =   2.18    1.835    0.999

b[]    1.0     131073.0     4.295E9     4.2949E9

roots =   2.088     1.915   0.9999

Thus the absolute values of the roots are 2.088, 1.915 and 0.9999.

Since f(-2.088) = 0, f(-1.915) = 0 and f(0.9999) = 0, the roots are -2.088, -1.915 and 0.9999

3. Find the root of   x3 - 4x2 + 5x - 2 = 0

a[]       1.0     -4.0     5.0     -2.0

b[]       1.0     6.0     9.0     4.0

roots =   2.449     1.2247     0.666

b[]      1.0     18.0     33.0     16.0

roots =   2.059   1.1636   0.834

b[]     1.0     258.0     513.0     256.0

roots =   2.002     1.0897    0.9167

b[]     1.0     65538.0     131073.0     65536.0

roots =   2.000    1.0443     0.9576

b[]     1.0     4.295E9     8.5899E9     4.295E9

roots =   2.00     1.0218     0.978

Thus the absolute values of the roots are 2.00, 1.0218 and 0.978.

Since f(2.00) = 0, f(1.0218) = 0 and f(0.978) = 0, the signs of the roots 2.00, 1.0128 and 0.978 are all positive.

4. Find the root of   x3 - 6x2  + 11x  - 6 = 0

a[]     1.0     -6.0     11.0     -6.0

b[]     1.0     14.0     49.0     36.0

roots =   3.74     1.87     0.857

b[]     1.0     98.0     1393.0     1296.0

roots =   3.146     1.942     0.982

b[]     1.0     6818.0     1686433.0     1679616.0

roots =    3.014     1.99     0.999

b[]     1.0     4.31E7     2.82E12     2.82E12

roots =   3.0002     1.999     0.9999

b[]     1.0     1.853E15     7.958E24     7.958E24

roots =  3.00     1.999     0.9999

Thus the absolute values of the roots are 3.00, 1.999 and 0.9999.

Since f(3.00) = 0, f(1.999) = 0 and f(0.9999) = 0, the signs of the roots 3.00, 1.999 and 0.9999 are all positive.

5. Find the root of    x3 - x2 - x + 1 = 0

a[]   1.0     -1.0     -1.0     1.0

b[]   1.0     3.0     3.0     1.0

roots =  1.73     1.0     0.577

b[]     1.0     3.0     3.0     1.0

roots =  1.316     1.0     0.759

b[]     1.0     3.0     3.0     1.0

roots =  1.147    1.0     0.8717

b[]     1.0     3.0     3.0     1.0

roots =  1.0711     1.0     0.9336

b[]     1.0     3.0     3.0     1.0

roots =  1.0349     1.0     0.9663

Thus the absolute values of the roots are 1.0349, 1.0 and 0.9663.

Since f(1.0349) = 0, f(-1.0) = 0 and f(0.9663) = 0, the roots are 1.0349, -1.0 and 0.9663.

6. Find the root of  x2 + 6x - 3 = 0

a[]       1.0     6.0     -3.0

b[]       1.0     42.0     9.0

roots = 6.481    0.463

b[]       1.0     1746.0     81.0

roots =  6.464     0.4641

b[]      1.0     3048354.0     6561.0

roots =  6.464     0.464

Thus the absolute values of the roots are 6.464 and 0.464.

Since f(-6.464) = 0 and f(0.464) = 0, the roots are -6.464 and 0.464

Problems to Work-Out:

 7. Find the root of  x4 - x - 4 = 0 8. Find the root of  2x3 - 3x2 + 2x - 3 = 0 9. Find the root of  x3- 5x2 + 4x - 3 = 0 10. Find the root of  x5 - x + 1 = 0 11. Find the root of  9x4 + 30x3 + 34x2 + 30x + 25 = 0 12. Find the root of  x5 - 2x4 + 4x3 - x2 - 7x + 5= 0