42. - = 4 3z 4x+y-2 3z :1 4 2x-y=2 x+y=49+2x2y a+y1-x2=20+ (2x2-1)y2. (x+y)3+23=1125; x+y+z=15; xy=24. 43. 44. 45. THEORY OF EQUATIONS. 1. Prove that a quadratic equation has two roots, and only two. 2. If a and ẞ are the roots of the equation ax2+bx+c=0, show that (1) a+b=−=; β and aß==; (2) that and are the roots of a3c2x2+(5a2bc2-5ab3c+b3)x+a2c3=0; (4) find the values of a+aß+ß3, a3+ß3, 3. Show that the roots of the equation x2+mx+n are real and different, real and equal, or impossible, according as m2>,=, or<4n. 4. If of the sum of the squares of the roots of the equation ax2+bx+c=0 is equal to their product, show that b2=5ac. 5. If a and ẞ are the roots of ax2+x+b=0, show that (1+)(1+)=ab. 6. If the equation x2+px+q=0 have equal roots, show that ax2+p(a+b)x+q(a+2b)=0 has one of them, and find the other. 7. Form the equation whose roots are 2, 1, −1. 8. When will the roots of x2+2px+q=0 be (1) both positive, (2) both negative? 9. If a, ẞ are the roots of ax2+bx+c=0, find in terms of a, b, and c, values of— 1 1 (1) +; (2) a3+ß3. 10. Show that impossible roots enter equations in pairs. What assumption is here made respecting the coefficients of the equations? In what case will irrational roots enter in pairs ? 11. Solve the equations (1) x3-7x-6=0, (2) 3x3+4x2-35x-12=0, which have a common root. 12. Form the rational equation two of whose roots are -1+3,-1-√3. 13. Reduce by 2 the roots of the equation x3-6x2+8x+1=0. 14. Find all the roots of the equation x4-2(x3+x2-x)+1=0. 15. Given x3-24x-72-0, to find x by Cardan's `method. 16. State and prove the relations between the coefficients and roots of the equation P+P-1 2-1+P2-2 2n−2+.. . P12+P=0. 17. Prove that the roots of the equation x3+prx—r2—() are the products of every pair of the roots of the equation x2+px2+r=0. 18. Show that the sum of the cubes of the roots of the equation x2+P1x2-1+P2xn−2+P3x-3+....=0 is 3p1p2-3p3—P13. 19. Transform the equation x3+px2+qx+r=0, whose roots are a, b, c, into one whose roots are ab, ac, bc. 20. Find all the roots of the equation x-6x2-16x+21=0. 21. Find the other roots of the equation xa — 2x3 — 6x2 +8x+8=0, one root being 1+√3. 22. 3+1 is a root of the equation x2-6x3+13x2-18x+30=0. Find all the other roots. 23. Find the sum of the fourth powers of the roots of the equation 4x3-7x2-x+6=0. 24. Prove that one real root of the equation x3-2x—5—0 lies between 2 and 2·1; also approximate to the value of the roots to 4 places of decimals. 25. Extract the cube root of 20, and find a root of the equation x3+2x=30 by Horner's method. 26. Solve the equation x3—12x=16, one root being 4. 27. Given x3-12x2 + 44x-48=0, and y=x-2; eliminate x and solve the resulting equation. 28. Show that 3—px2+qx−r=0 is the equation that results from the elimination of y and z from the equations x+y+z=p, xy+xz+yz=q, and xyz=r. 29. Solve the equation 4+4x3-18x2+20x-7=0, which has equal roots. 30. If the same transformation banishes both the second and third terms in the cubic equation x3-px2+qx—r=0, what relation must subsist between the coefficients ? 31. Prove that "—an is divisible by x- a; and show that if a be a root of the equation X=0, X is divisible by x-a. 32. Solve the equation x3=2x+2, by Cardan's method. CX. PROBLEMS. 1. An uncle is older than his nephew by 10 years; and 15 years ago the uncle was twice as old as his nephew. What are their respective ages? 2. Two shepherds, A and B, owning a flock of sheep, agree to divide it. A takes 144 sheep, and B takes 184 sheep, paying £70 to A. Required the value of a sheep. 3. A general, after losing a battle, found that he had only of his army fit for action; of the army were wounded, and the remainder, 2,000 men, were either killed or missing. Of how many men did his army consist at first? 4. There is a fraction which becomes equal to if 1 be added to its numerator, and becomes equal to if 1 be added to its denominator. Determine the fraction. 5. A workman was employed for 60 days, on condition that for every day he worked he should receive 2s. 6d., and for every day he was absent he should forfeit 10d.: at the end of the time he had to receive £2. How many days did he work? 6. A luggage train leaves a station, and travels at the rate of 10 miles an hour; after 4 hours another train follows from the same station, travelling 163 miles an hour. How far must the second train travel before it comes up with the first? 7. Find the fraction to the numerator of which, if 16 be added, the fraction becomes equal to 4, and if 11 be added to the denominator, the fraction becomes 4. 8. A man engages to perform, at a uniform rate, a certain distance on horseback in 10 hours, but when he is half way he increases his speed by 2 miles an hour, and thus arrives at his journey's end 14 hour before the time stipu lated. Required the distance he performed and the rate at which he started. 9. At the election of a member of parliament the successful candidate was returned by a majority of 88; but if 1 out of every 7 of his supporters had voted for his opponent, he would have been in a minority of 18. How many votes were recorded for each candidate? 10. A body of infantry is marching in regular column with 10 men more in depth than in front; on the enemy coming in sight the front is increased by 720 men, and then the infantry is drawn up in 9 lines. Find the number of men on the march. 11. Two casks, A and B, are filled with 2 kinds of sherry, mixed in A in the ratio of 2: 7, and in B in the ratio of 15. What quantity must be taken from each in order to have 11 gallons mixed in the ratio of 2: 9? 12. A has £3, and B has £2 8s. How much shall A give to B, that B may then have 3 times as much as A? 13. In a school of 600 children there are 18 more boys than girls, and the numbers of boys and girls are together double the number of infants. How many boys are there? 14. Two merchants, A and B, start with the same capital in trade; at the end of a year A has gained £3,000 and B has lost £1,000; it is then found that A's capital is double the capital of B. What did they each start with? 15. In a regiment 1,200 strong, the number of English and Scotch together was double the number of the Irish, and the number of English and Irish together was 5 times |