six times as much, or 62; 8x multiplied by x would give 8 x2, &c. If x multiplied by x gives x2, 3 x multiplied by x will give three times as much, or 3 x2; and 3 x multiplied by 4 x would give 12 x2, &c. 1. If x X x, or x.x, or xx, or x2, be divided by x, what will the quotient be? 2. If 6x2 be divided by x, what will the quotient be? 3. If 82 be divided by x, what will the quotient be?. 4. If 5x be divided by x, what will the quotient be? 5. Eight times a certain number is equal to twice its second power. What is the number? Let x= the number. Then 8x=2x2 Divide both numbers by x, and we have 82 x ; for if equal numbers be divided by the same number, or equal numbers, the quotients will be equal. Hence x 4. 6. Twice the second power of a certain number is equal to twenty-four times the number. What is the number? 7. There are two numbers, one of which is double the other, and their product is six times the sum of the numbers. What are the numbers? 8. There are two numbers, one of which is of the other; and the sum of their squares or second powers is equal to ten times the sum of the numbers. What are the numbers? 9. A merchant sold two pieces of cloth; of the number of yards in the first, is equal to the number of yards in the second. He sold each piece for as many shillings per yard, as the other contained yards. Both cost at this rate, six times as many shillings as there were yards in both pieces. many yards were there in each piece? How 10. Two men working together, built a piece of fence; one could build it all himself in 20 days; and the other could build it all himself in 30 days. Each man received as many shillings a rod as he built rods. And the whole came to twelve times as many shillings as there were rods of fence built. How many rods of fence were built? how many rods did each build? and how much money did each receive? 2x Let x= what one built; and what the other built. SECTION XV. Pure Equations of the Second Degree. The second root of a number, is that number which, multiplied by itself, will produce the number. It is sometimes called simply the root. 5 is the second root of 25; 6 of 36; 7 of 49; 4 of 16; 8 of 64; x of x2, &c. 1. Four times the second power of a number make 100. What is the number? 2. Five times the second power of a number make 500, What is the number? 3. One number is six times another, and the product of the two makes 216. What are the numbers? 4. The quotient of one number by another is 25, and their product is 100. What are the numbers? 5. There was a certain company of men; each man in the company had as many dollars as there were men in the company. They altogether had 625 dollars. How many men were there in the company? 6. One thousand dollars were paid at different times; at each payment ten times as many dollars were paid as there were payments. How many payments were there? 7. A number multiplied by its is equal to 18. What is the number? 8. The of a number multiplied by its, gives 24. What is the number? 9. One number is three times another. If the greater be multiplied by of the less, the product will be 36. What are the numbers? 13 10. If and of a certain number were added to the number, and the sum multiplied by the number, the product would be 264. What is the number? SECTION XVI. Exercises in Reducing Equations. Find the value of x in the following equations. 1. 6x-12=4x+16. х 2. 6x-4-2=x+1+30—19} 125 x 2 3. 12 18 x .40 4. Affected Equations of the Second Degree. 1. Multiply + 2 by x+2. It is evident that the product will be x times x + 2 added to twice x + 2. x2+4x+4 is, therefore, the second power of x + 2; and +2 the second, or square root of x2+4x+4. [SEE SECTION XV.] 2. Multiply +3 by x+3; x +4 by x+4; x + 6 by x+6; x+7 by x+7; x+8 by x+8; x+9 by x+9; x + 10 by x+'10; x + 12 by x+12; x + 11 by x + 11; x + 1 by x + 1, &c. 3. If x2+4x+4= 36; what is the value of x? The second root of one member of this equation, must be equal to the second root of the other; but the second root of x2+4x+4 is x + 2, and the second root of 36 is 6 Hence x+2=6 X= 4 4. If x2+6x+9=100; what is the value of x? 7. If x2+6x= 16; what is the value of x? Increase both members of the equation by 9, and we have, x2+6x+9=16+ 9, or 25. We can now take the second root of both members, and we have x+3=5 X= 2 8. What must you add to x2+8x, so that you may obtain the exact root of the sum? What to x2+10x? What to x2+12x? What to x2 + 14 x? What to x2+20x? What to x2+16x? What to x2+ 18 x?* Adding a number, so that you can obtain the root, is called completing the square. 9. If x2+8x= 20; what is the value of x? 10. The second power of a number added to ten times the number, makes 75. What is the number? 11. Says A to B, and if my number be product will be 300. I have 20 dollars more than you have; multiplied by the number you have, the How many dollars had each? 12. One number is 4 more than another, and their product is 21. What are the numbers? 13. One number is 6 more than another, and the product of the two is 160. What are the numbers? 14. There is a number, which being increased by 4, and * The second power of half the coefficient of x is the number to be added, also by 2; and the two sums multiplied together, the product Let x the number of yards the second digs. xXx = x2 the number of dollars the second received. x+8 multiplied by +8, gives x2+16x+64 = the number of dollars the first received, And both received 130 dollars; hence we shall have the following equation: Dividing both members of the equation by 2, we have x2+8x+32 = 65 x2+8x=33 Complete the square of the first member, and we have x2+8x+16= 49 x+4= x= x+8= 3 the number of rods the second built, = 11: the number of rods the first built, x2= 9 what the second received. x2+16x+64 = 121 = what the first received, Proof. 130 what both received, 16. A man sold two pieces of cloth, which differed in length by 4 yards. For each piece he received as many dollars per yard as there were yards in the piece, The whole sold for 136 dollars. How many yards were there in each piece? 17. A man had an orchard containing 341 trees. In the length of the orchard, there were 20 trees more than there were in the breadth. How many trees were there in the length, and how many were there in the breadth? SECTION XVIII. Affected Equations of the Second Degree. 1. Multiply x 2 by x The product will evidently be x times x-2, wanting twice x-2; but x times x-2 are x2 - 2 x, and twice x-2 are |