RULE.-Diminish the roots by 2 according to the method of 810. The coefficients of the transformed equation are A, B, C, D1. 4 is an approximation to the remainder of the root (832).

A1

B1

This gives .3 for the next figure of the root; but the highest figure must be taken which will not change the sign of A; this will be found to be .4.

Diminish the roots by .4. To do this, annex zeros to A, B, C, Dr as shown above, and use 4 instead of .4. Having found 4„, and noting that its sign is +, retrace the steps and try 5 instead of 4. This gives A, with a minus sign, hence the root lies between 2.4 and 2.5. The new coefficients are A, B, C2, D2.

next figure of the root.

gives 7 for the

Annex zero as before and diminish the roots by 1, representing the new coefficients by A, B, C, D.

The signs of A and B must remain unchanged.

If a change of

sign takes place it shows that too large a figure has been used.

834. The Abridgement of the Calculation. After a certain number of figures of the root have been found (say four), instead of annexing zeros, cut off one digit from B1, two from C, and three from D. This is equivalent to annexing the zeros and then dividing by 10000.

Continue this work with the numbers so reduced, and cut off digits in the same manner at each stage. until the D and C columns have disappeared.

Then A, and B, alone are left, and six more figures of the root are correctly determined by the division of A, by B.

The second root, which lies between 2.7 and 2.8, may be found in a similar manner.

1. Show that 2 is a root of 3— 7x '+ 6 = 0.

2. Show that 3 is a root of 2x3 + 5x2 + 9 = 0.

3. 1 is a root of x1 3.r2+4x 2 = 0; find the others.

5. Solve the equation + 2x3-5x+6x+2= 0, which has a

8. Transform the equation 12.5 34x2+33x-1=0 into another which shall have the same roots with opposite signs.

3.5 + 7.5

1.250 into

10. Transform the equation a3 another whose roots are double those of the given equation. 11. Transform the equation 3 12.2 another whose roots are of the roots of the given equation.

12. Solve the equation roots are in the ratio of 3 to 2.

18x + 135 = 0 into

9x2 + 14x + 24 = 0, two of whose

14. Solve the equation 27.3 + 42x2 — 28.x are in G. P.

=

15. Transform the equation a3 a.x2 bx c 0 into another whose roots are the square of the roots of the given equation.

16. The equation 3. – 25.3 + 50.x2

roots whose product is 2; find all the roots.

50x120 has two

17. Show that the equation æ3. c2 10 has one real root

only.

18. Show that

2.c20 has a real negative root.

19. Show that x1 + 2x3 + x2 + x − 1 = 0 has two real roots.

20. Discuss the roots of + 2.x3 — x2 — 1 = 0.

21. Find the inferior limit to the number of imaginary roots of the equation axo − 3x2 − x + 1 = 0.

-

22. Find the nature of the roots of the equation a1 + 15x2 + 7x - 11 = 0.

23. Find the multiple roots of the equations:

24.

Determine the number and situation of the real roots of the

(f)

20x47 = 0.

x2 + 8.x3-30x210.x + 241 = 0.

(g) x7x+33x2-55x+80= 0.

Determine the real roots of the following equations by Horner's method:

28. 2.x3 12x2 + 9x + 24 = 0. Ans. 4.3098; 2.7155; -1.0253.

29. The equation 2.3

between 300 and 400; find it.

650.8r + 5 - 16270 has a root Ans. Commensurable root 325.4.

30. Find the root between 20 and 30 of the equation

4x3 - 180x2 + 1896x 457 0.

33. 2x4x3 + 3.x2 - 1 = 0.

34. 3.x1 2.x3 212

4x+11= 0.

= 0.

Ans. 28.52127738.

Ans. 2.8809; — 2.8193.

Ans. 3.7509; -3.8048.
Ans. +1; 0.4406.

Ans. 4.0071; 0.6339;0.9503;2.0241. 15x+2x+31

Ans. 1.5055; 0.5367;0.5397;0.8025. + 2x + 3x3 + 4x2 + 5x = 321 has one real Ans. 2.638605803327.

Equations, numerical and literal 173 Equations, (continued):