QUADRATIC EQUATIONS.] 



MATHEMATICS. ALGEBRA. 



475 



.-.Rule II., 



= V3G1=+19 



.. ,.. 



4 2 



so that the values of x and y are either 



= 91 



y=3J or }y = 

 Sometimes pairs of equations, coming under the present 

 head, may be most conveniently solved, independently 

 of rule, by exercising a little ingenuity. Here is an 

 example : 



3. ,r+y=121 By squaring the first equation, and then sub- 

 *- + y - = 74 J trading the second, 2.ry = 70 . . 4 .ry = 140. 

 From the sq. of the first, viz. x" + 2.ry +y 2 = 144 

 Subtract 4jry =140 



Extracting the square root, 

 But, 



x-y 



-= 4 



= +2 

 = 12 



. . adding and subtracting, Zx = 12 + 2, and 2y = 1 2 + 2 



. . x=."l, or 5 ; and y=5, or 7 

 that is, the values of x and y are either 

 * = 71 ,, fy = 7 

 y = 5j or ti= 



4. In a similar vray you may treat the equations x y = 6, 

 * : -if y* = 50 ; and you will find that the values of x and y are 

 either x = 7, y=l; orx = liy = 7. 



lions requiring the Solution of Quadratic Equations. 

 1. The fore-wheel of a carriage makes C revolutions 

 more than the hind-wheel in going 120 yards ; but it is 

 found that if tho circumference of each wheel be increased 

 1 yard, it wi'l make only 4 revolutions more than the 

 hind-wheel in the same distance ; required the circum- 

 ference of each wheel. 



Let x = No. of yards in circ. of larger wheel, 

 y smaller wheel, 



then = No. of revolutions of the former, and = 

 x y 



120 



No. of revolutions of latter ; and by the question, =: 



120 



6.-.20y = 20x xy.-. *i/ = 20.c 20y. . [A]. 



Also by the question, 3^-= -^ 4 . . 30(y + 1 



x -p -i y ~T~ J- 



= (z + 1) (29 y), that is, 30y + 30 = 29* + 29 xy 

 y.-. xy = 29x 31y L 



Substituting this expression for xy in the equation [A]> 

 we have 



29.e Sly l = 20x 20i/. . 9x = llt/ + l. . . . [B]. 

 This equation [B] Is a simple equation, and [A] is a 

 quadratic, or an equation of two dimensions, because of 

 the term xy. 



From [A] wehavejr = ^i i this substituted in [B] 



glVi 



180y 



= 40y + 20 .-. lly' 39y = 20 

 .-. Rule II., 22y 39 = V (880 + 39 1 ) = / 2401 = + 49. 



39 + 49_ 5 . _lly+l_ ' i 



From the nature of the question, it is plain that the 

 negative values of x and y are inadmissible ; they fulfil 

 the algebraical conditions [A] and [B], as well as the 

 positive values, for there is nothing in those conditions to 

 exclude them ; and it will often be found that tho 

 algebraic translation is free from the particular restric- 

 tions embodied in the question itself. The algebra 

 furnishes all tho values of the symbols, whether positive 

 or negative, real or imaginary ; and those of them are 

 afterwards to bo rejected which the restrictions of tho 

 question necessarily exclude. In the present case tho 

 only answers to the question are x = 5, and y = 4 



2. A company at a tavern had S 15s. to pay ; but 

 before the bill was settled, tvro of them left ; in conse- 

 quence of which, those who remained had each 10s. more 

 to pay. How many persons were in company at first ? 



Let x represent the number : then - is the share of 



each in shillings, and 



175 



-^ 

 x 



two had left : the difference is 



portion each paid, after 

 - = 



-- = 10 by tho 



question, . . clearing fractions, 



175x 175* + 350 = 10x(x 2) ; 

 or, dividing by 10, 35 = x- 2x, or x 2 2x = 35, 



.'. 2x 2 = 



= 1+6=7, or 5. 

 Consequently they were 7 persons at first. 



3. What number is that wliich, being divided by the 

 product of its two digits, the quotient is 2 ; and if 27 

 be added to it, the digits will be inverted or transposed ? 



Let x and y be the digits, then the number is 10* + y 

 and when the digits are inverted or transposed, the 

 number is lOy + x. 



By the question, -^-=2 .'. 10j? + y=iry . . . . [A] 



and 



Substituting this value of y in [A], we have 



1 Ix + 3 = 2a s + Gx .-. 2x'5x = 3 

 .-. 4.r 5 = v' (24 + 25) = V 49 = + 7 . 



x= == = 3, or J.'. y = * + 3 = 6, or 2}. 



The only admissible values are jr = 3, and y = 6, .'. the num- 

 ber is 36. 



J. Divide tho number 33 into two such parts, that 

 their product may be 162. 



2. Find two numbers whose difference is 9, and which 

 are also such that their sum multiplied by tho greater 

 gives 266 for the product. 



3. A company at a tavern had 7 4. to pay ; but two 

 of them having left, the others had each Is. more to pay 

 than his fair share ; how many persons were there at 

 first? 



4. A purse contains 24 coins of silver and copper ; 

 each copper coin is worth as many pence as there are 

 silver coins, and each silver coin is worth as many pence 

 as there are copper coins : and the whole is worth 18s. 

 How many are there of each ? 



5. Two messengers, A and B, were dispatched to the 

 same place, 90 miles distant. A, by riding one mile an 

 hour more than B, arrives at his destination an hour 

 before him. How many miles an hour did each travel ? 



6. A grocer sold SOlbs. of mace and lOOlbs. of cloves 

 for 05 ; but he sold 601bs. more of cloves for 20 than he 

 did of mace for 10. What was the price of lib. of 

 each? 



7. The product of two numbers is 240, and they are 

 such, that if one of them be increased by 4, and the 

 other diminished by 3, tho product of the results ia still 

 240. Find the numbers. 



8. A and B set out at the same time for a place 150 

 miles distant. A travels 3 miles an hour faster than B, 

 and arrives at the place 8 3 hours before him. How 

 many miles did each travel per hour ? 



9. What number is that the sum of whose digits is 15, 

 and if 31 be added to thtir product, the digits will be 

 transposed ? 



10. There is a certain number consisting of two digits. 

 The left-hand digit is equal to three times the right-hand 

 one ; and if 12 be subtracted from the number, the re- 

 mainder will be equal to the square of the left-hand 

 di"it. What ia the number ? 



A pair of simultaneous equations, with two unknown 

 quantities, cannot in general be solved without the aid 



