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. 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. PAGE Equations, numerical and literal 173 Equations, (continued): PAGE problems leading to . 180 factors of 404 infinite, solution of. of the first degree 198 simultaneous system of linear 198 concerning the theorem of Pythagoras 412 indeterminate 198 concerning the area of plane figures. 418 has two roots only 422 relation between the roots and properties of roots of Equations which are biquadratic 427 roots of biquadratic a+b+c=0,427 solution of ax2+bx2+c=0. 430 Equations which are irrational 428 solution of ax2+bx+2lv ax2+bx+c=p. 430 Equations of the form ax2+bx2+c=0 Equations which are called recip rocal solution of 430 431, 761 432 Equations, simultaneous quadratic in two unknown quantities 451 451 453 456 .457 464 232 233 type I résumé of discussion . type II 236 homogeneous equations type III 236 two equations which have com irrational quadratics in three unknown quantities special methods of solution 464-470 graph of y=ar+ bx + c 241 Equation, the cubic rule for solution of three equations Equations, n linear equations of the first degree, solution of . 242 problems involving three or more linear equations . 251 graphs of solution, see "Graph" 265 Equations, diaphantian equations and problems indeterminate equations of the first degree indeterminate equation ar+by=c,268 indeterminate equation ax-by=c,272 general solution of two indeterminate equations type I type II a root of cube roots of unity resolvent cubic. Equations, theory of properties, 1-6 transformations of 480 488 489 491 736 736 737 738 .739 268 268 741 . 742 745 .746 757 757-761 274 761-767 283 283 Equations of the Second Degree. 385 introduction, theorems I -V, 385-387 solution of pure Quadratic arb ax2+px+y=0 392 405 398, 771 factors of roots of equal roots. imaginary roots real and different real and equal imaginary and unequal solution of +px+q Evolution, definition of a root the radical sign, radicand |