16 And x+y+z=-+4+z=14, by the 2d equation, Z Or 16+42+z? =142, or 22 - 10z=- 16, Whence z=5+/25-16=5+3=8, or 3 by the rule, Therefore the three numbers are 2, 4, and 8. 9. It is required to find two numbers, such that their sum shall be 13(a), and the sum of their fourth powers 4721 (6). Let x= the difference of the two numbers sought, 1 atx Then will X, = the greater number, 2 2 2 at or 16 (a+x)" + (a—x)" But =b, by the question, 16 Or (a+x)4 +(2-x)4=166, by multiplication, Or 224 +12a2x2 +224=16b, by involution and addition, And w* +6a2x2=8b-a“, by transposition and division, Whence x3 = -30+994 +86-a4=-3a + ✓8(a++b), by the rule, And x= -3a2 +22(a+ +b), by extracting the root, Where, by substituting 13 for a, and 4721 for b, we shall have x=3, 13+ 16 Therefore =8, the greater number, 2 2 10 13. And =5, the less number, 2 2 | The sum of which is 13, and 84+54-=4721. QUESTIONS FOR PRACTICE, 1. It is required to divide the number 40 into two such parts, that the sum of their squares shall be 818. Ans. 23 and 17 2. To find a number such, that if you subtract it from 10, and then multiply the remainder by the number itself, the product shall be 21. Ans. 7 or 3. 3. It is required to divide the number 24 into two such parts, that their product shall be equal to 35 times their difference. Ans. 10 and 14 4. It is required to divide a line, of 20 inches in length, into two such parts that the rectangle of the whole and one of the parts shall be equal to the square of the other. Ans. 105 - 10 5. It is required to divide the number 60 into two such parts, that their product shall be to the sum of their squares in the ratio of 2 to 5. Ans. 20 and 40 6. It is required to divide the number 146 into such two parts, that the difference of their square roots shall be 6. Ans. 25 and 121 7. What two numbers are those whose. sum is. 20 and their product 36 ? Ans. 2 and 18 8. The sum of two numbers is 1}, and the sum of their reciprocals 3 ; required the numbers. Ans. į and 9. The difference of two numbers is 15, and half their product is equal to the cube of the less number; required the numbers. Ans. 3 and 18 10. The difference of two numbers is 5, and the difference of their cubes 1685 ; required the numbers. Ans. 8 and 13 11. A person bought cloth for 331. 158. which he sold again at 21. 8s. per piece, and gained by the bargain as much as one piece cost him ; required the number of pieces. Aps. 15 M 12. What two numbers are those, whose sum, multiplied by the greater, is equal to 77, and whose difference, multiplied by the less, is equal to 12. Ans. 4 and 7 13. A grazier bought as many sheep as cost him 60l., and after reserving 15 out of the number, sold the remainder for 541., and gained 23. a head by them : how many sheep did he buy? Ans. 75 14. It is required to find two numbers, such that their product shall be equal to the difference of their squares, and the sum of their squares equal to the difference of their cubes. Ans. iv 5 and 1(5+5) 15. The difference of two numbers is 8, and the difference of their fourth powers is 14560 ; required the numbers. Ans. 3 and 11 16. A company at a tavern had 8l. 15s. to pay for their reckoning ; but, before the bill was settled, two of them wept away ; in consequence of which those who remained had 10s. apiece more to pay than before : how many were there in company ? Ans. 7 17. A person ordered 71. 4s. to be distributed among some poor people ; but, before the money was divided, therè came in, unexpectedly, two claimants more, by which means the former received a shilling a piece less than they would otherwise have done ; what was their number at first? Ans. 16 persons 18. It is required to find four numbers in geometri. cal progression such, that their sum shall be 15, and the sum of their squares 85. Ans. 1, 2, 4, and 8 19. The sum of two numbers is 11, and the sum of their fifth powers is 17831 ; required the numbers ? Ans. 4 and 7 20. It is required to find four numbers in arithmetical progression such, that their common difference shall be 4, and their continued product 176985. Ans. 15, 19, 23, and 27 21. Two detachments of foot being ordered to a station at the distance of 39 miles from their present quarters, begin their march at the same time; but one party, by travelling of a mile an hour faster than the other, arrive there an hour sooner ; required their rates of marching ? Ans. 31 and 3 miles per hour 22. It is required to find two numbers, such that the square of the first plus their product, shall be 140, and the square of the second minus their product 78. Ans. n and 13 OF CUBIC EQUATIONS. A cubic equation is that in which the unknown quantity rises to three dimensions ; and like quadratics, or those of the higher orders, is either simple or compound. A simple cubic equation is of the form 6 b ax3=b, or x3= ; where x=; = a A compound cubic equation is of the form x3 + ax=b, x3 tax?=b, or x3 tax2 + bx EC, in each of which, the known quantities a, b, c, may be either + or Or, either of the two latter of these equations may be reduced to the same form as the first, by taking away its second term ; which is done as follows : RULE. Take some new unknown quantity, and subjoin to it a third part of the coefficient of the second term of the equation with its sign changed ; then if this sum or dif. ference, as it inay happen to be, be substituted for the original unknown quantity and its powers, in the preposed equation, there will arise an equation wanting its second term. Note. The second term of any of the higher orders of equations may also be exterminated in a similar manner, by substituting for the unknown quantity some other unknown quantity, and the 4th, 5th, &c. part of the coefficient of its second term, with the sign changed, according as the equation is of the 4th, 5th, &c. power. EXAMPLES. Here x= 1. It is required to exterminate the second term of the equation 23 +3ax?=b, or x3 +30x2_b=0. За -a, 3 x3=23 - 3uż2 + 3a22-a3 Then 3ax2 = +3az? – 6a2zt3a3 b= -b Whence 23 - 3a’zt2a3-b=0, Or 23 -8a2z=b-2a3, in which equation the second power (22), of the unknown quantity, is wanting. 2. Let the equation 23 – 12x2 + 3x=-16, be transformed into another, that shall want the second term. Here x=x+4, (2+4)3=23 +1222 +482 +64 Then -122+4)2= 12z2 – 962-192 +3(2+4) +32 +12 Whence z3 - 452-1163-16 Or 23-452=100 which is an equation where 2%, or the second term, is wanting, as before. |