Ex. 41. Solve the equations— 5. x3-6x2+10x-8=0. 6. 27x3-135x2+225x-117=0. 7. x2-27x+25=7x2(5-x). 8. x-2x3+x=132. 9. x+x3-4x2+x+1=0. 10. x—x3 + 1⁄2 x2−x+1=0. [N.B. x(x2-6x+8)+2(x−4)=0.] [N.B. 3-3 is a divisor.] [N.B. (x−5)2—7x3 (x−5)++2x=9x. [N.B. x+—2x3+x2 — (x2—x) = 132.] 11. xa—8x3+10x2+24x+5=0. [N.B. (x2—4x)2—6(x2−4x)+5=0.] 12. x++13x3-39x=81. [N.B. x2-81+13(x2—9)x=0.] VI. SIMULTANEOUS EQUATIONS OF THE 2ND, 3RD, &c. DEGREES. Ex. 42. Solve the equations 49. x2(b-y)=ay(y-n), ] 50. (x2 —xy+y2)(x2+y2)=221, y' (a-x)=bx(x-n).S (x2-xy + y2)(x2 + xy + y2)=273.5 VII. PROBLEMS IN EQUATIONS OF THE 2ND AND HIGHER DEGREES. Ex. 43. 1. Find two numbers, whose difference is two-ninths of the greater, and the difference of whose squares is 128. 2. The sum of two numbers is 16; and the quotient of the greater divided by the less is 23 times the quotient of the less by the greater: find them. 3. The difference of two numbers is 15, and half their product is equal to the cube of the less number: find them. 4. The product of two numbers is 24, and their sum multiplied by their difference is 20: find them. 5. The difference of the squares of two consecutive numbers is 17: find them. 6. The product of two numbers is 18 times their difference, and the sum of their squares is 117: : find them. 7. What two numbers are those whose sum multiplied by the greater is 204; and whose difference multiplied by the less is 35? 8. There are two numbers such that the sum of the products of the first multiplied by 4 and of the second by 3 is 53; the difference of their squares is 15: find the numbers. 9. The product of two numbers added to their sum is 23; and 5 times their sum taken from the sum of their squares leaves 8: required the numbers. 10. Divide the number 14 into two parts, such that the sum of the quotients of the greater divided by the less, and of the less by the greater may be 22. 11. What two numbers are those whose sum added to the sum of their squares is 42, and whose product is 15? 12. A farmer bought some sheep for £72, and found that if he had received 6 more for the same money, he would have paid £1 less for each. How many sheep did he buy? 13. A and B distribute £60 each among a certain number of persons: A relieves 40 persons more than B does, and B gives to each 5s. more than A. How many persons did A and B respectively relieve? 14. A vintner sold 7 dozen of sherry and 12 dozen of claret for £50. He sold of sherry 3 dozen more for £10 than he did of claret for £6. Required the price of each. 15. A detachment from an army was marching in regular column, with 5 men more in depth than in front; but upon the Ex. 43. enemy coming in sight, the front was increased by 845 men; and by this movement the detachment was drawn up in 5 lines. Required the number of men. 16. The product of the sum and difference of the hypothenuse and a side of a right-angled triangle is 2; and 4 times the sum of the squares of the hypothenuse and this side is equal to 5 times the sum of these two lines: find the 3 sides of the triangle. 17. There are three numbers, the difference of whose differences is 5; their sum is 44, and continued product 1950: find the numbers. 18. Divide the number 26 into three such parts that their squares may have equal differences, and that the sum of those squares may be 300. 19. The sum of 4 numbers is 44; the sum of the products of the first and second, and third and fourth is 250; of the first and third, and second and fourth is 234; and of the first and fourth, and second and third 225: find them. 2. If x2=a2+b2, y2=c2+d2; show that xy>ac+bd or ad+bc. 3. If ab; show that a-b>(abi)2. —b>(až 4. If x>y; show that x- -y>· -2 5. Show that ay2+x-2y>x-1+y ̃1; x2+ y2+z3>xy +xz+yz. 6. Show that 2(1+a2+a4)>3(a+a3). 7. If 4b>a2; show that a2+b2>ax. 8. If x>a; show that x3+7ax2>(x+a)3. 9. Show that (a+b+c)3>27abc, but <9 (a3 +b3 + c3). 10. Show that abc>(a+b−c) (a+c—b) (b+c-a). 11. Show that 3+y3+z3>±(x2y+xy2+x2z+xz2+y2z+yz2). Ex. 45. RATIO, PROPORTION AND VARIATION. 1. Compare the ratios 7 : 8 and 10: 11; 19: 25 and 56 : 74. 2. Show that a: b>ax: bx+h; but <ax: bx-h. 3. Show that a3 + b3 : a2+b2>a2+b2: a+b. 4. Which is greater, a+x: a−x, or a2+x2 : a2 — x2 ? |