x2 (2.) 3x2 — — 54 — 2x2. Ans. 12=2√3. . x=±√6. x=+2. x=3, or 1. (25.) 5(x2 — 1) — 4(3x-2)+ 1 = 0. x=2, or 3. (33.) 3(2−x) + 2(3—x)=2(4+3x2). x = 1⁄2, or — 1§. (34.) x2+1+(x+1)2 = (x + 2) (x+3). x = 4, or — 1. (38.) 5x2 -5(6- x)2 = 24x(6 — x).. x=5, or — §. (43.) (x + 3)3 + (x − 3)* — (3x+3)1. x = 5, or — 3. = 82 (44.) 2√(3x-5)=3+ √(5x-1). . x=10, or §. (45.) √(x−1)(x—2)+√(x−3)(x−4)=√2. x = 2, or 3. (46.) x2-√(x2-7)= 13. . . . . x=4, &c. (47.) x2—x—3,√(3x2-4x+1)=-7. x= 3, or—§,&c. (48.) 5x-3x=14. (49.) x2 = 31+ √(x2 — 11). (50.) 2x-(2x+6)= 6. x= 4, or 18. x = 5, or 3. (51.) 3x2+2√(2x2—3x+7)=x2+3x+17. x = 3, or—§,&c. (52.) 9x-4x2+ √(4x2 — 9x+11) = 5. x=2, or 4, &c. Ans. (53.) 3x(3—x)=11—4√(x2 — 3x+5). x=4(3±√5),&c. a + x (59.) + ax (a—x) √(a + x) = (b X= 12a 5 √(a + x)° x=3a, or 2a. x = a, or 3a. =2√a. x=±a(±8,√2—11)*. (60.) x2+2a√(x2 + ax—a2)=4a2—ax. x=a, or—2a, &c. 骤 (61.) 2(x2 — ax) — √/ ( x 2 — ax — a 2 ) = a(4a — 1). 2 2 (62.) (~-—1)2 + („,. + 1)2 = n(n − 1). x=2a, or-a,&c. (63.) x2+y2 = 34, and xy = 15. x=±5; y =±3. (64.) x2-y2=27, and xy-y2-9. x = ±6; y = ±3. (65.) x2+y2=20, and x+y=6. x = 4, or 2; y = 2, or 4. Ans. , or 5. (70.) x3+y3=133, and x+y=7. x=5, or 2; y = 2, (71.) x3—y3=117, and x-y=3. x=5,or—2; y=2, or—5. (72.) x+y=1297, and ry=6. (73.) x4-y1-240, and x-y=2. x=6,&c.; y=±1,&c. x=4, &c.; y=2, &c. (74.) x+y=a, x2+z2 = b, and y2+22= c. x= ±(a+b—c) * ; y = ±("-—-—b+c)" ; ; 2 = a+b+c QUESTIONS PRODUCING QUADRATIC EQUATIONS. (1.) Find a number which is less than its square by 6. Ans. 3, or (2.) What is the number, to which if its square and cube be added, the sum is equal to thirteen times the number itself? Ans. 0, 3, or - 4. (3.) Find two square fields having the sum of their areas an acre, and the difference of their sides 8 perches. Ans. The sides are 4 and 12 perches. (4.) Find a number such that if it be increased by 5, and diminished by 2, the product of the results may be 60. Ans. 7, or - 10. (5.) Find a number such that if it be diminished by 5, and increased by 2, the product of the results may be 60. Ans. 10, or -7. (6.) Required a number such that if 5 be taken from its double, and 11 from its treble, the product of the remainders may be 20. Ans. 5, or 1. (7.) Required a number such that if 2a be added to it, and 26 subtracted from it, the product of the results may be 3a+2ab. Ans. a 2b, or —3a. (8.) Find a number such that if 12 be subtracted from its double, and the remainder multiplied by the number itself, the product shall be 14. Ans. 7, or - 1. (9.) Find a number such that if it be multiplied by n, and a subtracted from the product, the remainder, multiplied by n times the number itself, shall be equal to b2. (10.) Divide 20 into two such parts that the sum of their squares shall be 208. Ans. 12, and 8. (11.) Divide 210 into two such parts that the one is the square of the other. Ans. 14, and 196. (12.) Divide a into two parts such that the one is the square of the other. 552. Ans. 1(1+4a) 2 (13.) Find two consecutive numbers whose product is Ans. 23, and 24. (14.) Divide 18 into two such parts that twice the square of the greater shall exceed three times the square of the less by 95. Ans. 11, and 7. (15.) Divide 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. (16.) Divide 225 into two such parts that the sum of their square roots shall be 21. Ans. 144, and 81. (17.) Find two numbers whose difference is, and the difference of their reciprocals. - • Ans., and; or —, and (18.) Find two numbers whose sum is 52, and whose product is three times the cube of the less. Ans. 4, and 48; or 4, and 56. (19.) Find two numbers, each consisting of two digits, and such that their difference shall be twice the unit figure of the greater, their sum five times the product of the digits of the greater, and their product 384. Ans. 16, and 24. (20.) A grazier bought a number of sheep for £40, 16s., and sold them again at £1, 15s. 6d. per head; he gained 2s. more than one sheep cost him; how many did he buy? Ans. 24. (21.) A grazier bought a number of sheep for £60; reserving 15, he sold the remainder for £54, and gained 2s. per head; how many did he buy? Ans. 75. (22.) Find four consecutive numbers such that if the first two be taken as the digits of a number, that number is the product of the other two. Ans. 1, 2, 3, 4; or 5, 6, 7, 8. (23.) Find three consecutive numbers such that if the last be multiplied by the sum of the other two, the product is equal to the number expressed by the first and last taken as digits. Ans. 3, 4, and 5; or 0, 1, and 2. (24.) A person bought a number of sheep for £150; and |