Whence z=30=7 (900— 864) =305/36=30+6, by the rule, And consequently x=30+6=36, or 30—6=24, the two parts sought. 3. It is required to find two numbers such that their sum shall be 10(a), and the sum of their squares 586). Let x=the greater of the two numbers, Then will a-r=the less, And x2 +(a-x)=2x2 - 2ax+a=b, by the question, Or 2x2 - 2ax=b-a", by transposition, b-a? by division. 1 26-a? by the rule, 10. 1 + 101 And 10—3= ē - 116–100=3, the less, ax= a a 1 yalue of x. 16. Given vite–23–2(1+x=x2)=, to find the . == v(x-7)+r(1-2=r. 1 1 Ans. x=-tav41 2 of x. 17. Given v to find the value Ans. x=1 +\v 5 18. Given xen — 2.03ni+x"=6, to find the value of x. Ang. x=+iv 13 QUESTIONS PRODUCING QUADRATIC EQUATIONS. The methods of expressing the conditions of questions of this kind, and the consequent reduction of them, till they are brought to a quadratic equation, involving only one unknown quantity and its square, are the same as those already given for simple equations. 1. To find two numbers such that their difference shall be 8, and their product 240. Let x equal the least number. Then will x+= the greater, And x(x+8)=ra +8x=240, by the question, Whence r=-4+ 16+240=-4+256, by the com mon rule, before given, Therefore x=16.4=12, the less number, And x t-8=12+8=20, the greater. 2. It is required to divide the number 60 into two such parts, that their product sball be 864. Let x=the greater part, Then will 60 - x=the less, And x(60- x)=50x - x2=864, by the question, Or by changing the signs on both sides of the equation x? - 60x= -864, Whence x=30+7 (900-864)=305/36=30+6, by the rule, And consequently x=30+6=36, or 30-6=24, the two parts sought 3. It is required to find two numbers, such that their sum shall be 10(a), and the sum of their squares 58(b). Let x=the greater of the two numbers, Then will a-(=the less, And x2 +(a – »)=2x2 - 2ax+ao=b, by the question, Or 2.2 - 2ax=b-a?, by transposition, bwa? And x2 , by division. 2 ar = b 1 Whence x= a2 4 2 by the rule, And if 10 be put for a, and 58 for b, we shall have 10 1 4. Having sold a piece of cloth for 241., I gained as much per cent. as it cost me; what was the price of the cloth ? Let x= pounds the cloth cost, Then will 24-2=the whole gain, That is, xa +100x=2400, by the rule, 5. A person bought a number of sheep for 801., and if he had bought 4 more for the same money, he would have paid 1l. less for each ; how many did he buy? Let x represent the number of sheep, 80 80 80 But +1, by the question, 80x x+4 And consequently x=16, the number of sheep. 6. It is required to find two numbers, such that their sum, product, and difference of their squares, shall be all equal to each other. Let x=the greater number, and y= the less. Then { 2+y="_y:}by the question. х2 – уг x+y 1 1 Whence y=+v6+1)= + v5, by the rule, Therefore y=+v5=1.6180 ... 3 1 2 Where ... denotes that the decimal does not end. 7. It is required to find four numbers in arithmetical progression, such that the product of the two extremes shall be 45, and the product of the means 77. Let x= least extreme, and y=common difference, Then x, x+y, x+2y, and x+3y, will be the four num bers, Hence : S x(x+3y)=+3xy=45 tion, x+2y)= 32 = 16 by division, and y=v16=4, 2 Therefore x3 + 3xy=x2 +12x=45, by the 1st equation, And consequently x=-6+ ✓ (36 + 45) = -6+9=3, by the rule, x+y(2429)=x2+3y+-242=77} by the ques Or ya = Rate TE64} by the question , 8. It is required to find three numbers in geometrical progression, such that their sum shall be 14, and the sum of their squares 84. Let x, y, and z be the three numbers, 5x+y+z=14 + y2 +=84 Hence xtz=14-y, by the second equation, And 22 +223+2:=196 - 28y+yo, by squaring both sides, its equal 2x2, Or 196 - 28y=84 by equality, 196--84 Hence y =4, by transposition and division, 28 Again &z=y2=16, or r=- by the 1st equation, |