2x -4. Subtract 2 x 4 from x2. 2x-2x+4= x2. -2x, and we have x2. 4x+4. We shall obtain this result by multiplying, and placing the sign + before the product of each two terms which have like signs; and before the product of each two terms which have different signs. Thus X -2 4. If x2 5. If x2. 2? What is the sec •4x+4= 100; what is the value of x? 12x+36=144; what is the value of x? 6. Eight times a certain number, subtracted from its second power, leaves 9. What is the number? 7. If 6 be subtracted from a number, and the remainder be multiplied by the number, the product will be 40. What is the number? 8. Says A to B, you have twice as much money as I have, wanting 12 dollars; and if the number I have be multiplied by the number you have, the product will be equal to 80. How much money have A and B ? 9. A and B have equal sums of money. A lost 7 dollars, and B 5. Then the number of dollars A had, multiplied by the number B had, will be equal to 99. How much money had A and B ? 10. There are two numbers, one is three times as large as the other, wanting 6, and the product of the two is 24. What are the numbers? 11. The sum of two numbers is 20, and their product 96. What are the numbers? In this equation a2 is, or negative; it is necessary before. we can apply the process of solution which has been explained, to make x2+, or positive. Increasing both members of the equation by x2, we have 20 x 96 x2 Subtracting 96 from both members, we have 20 x 96 x2 Subtracting 20 x from both members, and we have Subtract 2 from 6 and 4 remains. Now suppose we have 6 to subtract, and only 2 to subtract from. We can subtract 2 of the 6 from 2, and there is nothing left; but there are still 4 more of the 6 to take away, and nothing to take it from; we write it 0-4, or simply —4, which shows that the 4 is to be subtracted: 6-2=+4, but 2-6=-4. Upon the same principle, if 20 x- x2+96, Then x2-20 x =- -96 12. The sum of two numbers is 16, and their product 60. What are the numbers? 13. The sum of two numbers is 50, and their product 600, What are the numbers? 14. The sum of two numbers is 36, and their product 320, What are the numbers? 15. The sum of two numbers is 40; if their difference be multiplied by the less number, the product will be 72. are the numbers? 16. The sum of two numbers is 20; if what one wants of 12 be multiplied by the excess of the other over 12, the product will be 45. What are the numbers? SECTION XIX. Affected Equations of the Second Degree. 1. Multiply x + 1 by x + 1? x + х 2. Multiply x + by x + 1; x + 1 by x + 1; x + by x + 3 ; x + 2 by x + 2; x + by x + 1; x + by x+}; and x+by+3. 3. If x2+x+1=61; what is the value of x? 5. If x2+=+= 5; what is the value of x? 2 6. If x2+x= 6; what is the value of x? 7. If x2+ =53; what is the value of x? 2x 8. If from a certain number be subtracted and the remainder be multiplied by the number, the product will be 5. What is the number? 9. There are two numbers which, added together, make 10; but if one be increased by, and the other be diminished by , the product will be 243. What are the numbers? 10. Seven hundred and fifty men are drawn up in a solid body of an oblong form; the number of men in rank added to the number in file would be 55. How many were there in rank, and how many in file? 11. If 25 be divided by a certain number, the quotient will be 2 more than if 45 were divided by a number greater by 10. What is the number? SECTION XX. Affected Equations of the Second Degree. 1. What is the second power of x-5? Ans. 10x+25. 2. What is the second power of 5-x? Ans. 25-10x + x2. 25 has, We see that the second power of x- 5 and the second power of 5- x are precisely the same. x2-10 x therefore, two roots, x 5 and 5-x. then x 5 would be + 2, but 5 x2-10x+25 would be +4. Suppose x to be 7, x would be - 2. .3; 4 has therefore two roots, +2 and -2. If x were 8, then x 5 would be +3, and 5-x would be 10x+25 would be +9; 9 has, therefore, either +3, or 3 for its root. What then would be the root of -9? It cannot have a root, since the second power of either +3 or -3 would be 9. A negative quantity has, therefore, no second root; and every positive number has two roots; one plus and the other minus. 3. Find two numbers whose sum is 15, and whose product is 50. By making x one of the numbers we shall obtain the following equation: The root of the first member is x -7, and the root of the second member, is either +2 or the root, and we have x-7=21, then x = = 10; but suppose -21 to be the root of 61, and we have x-7=-21, Then x= 71—21, or = = 5. We have, then, two answers, 10 and 5. 4. The sum of two numbers is 80, and their product 1200. What are the numbers? Find the two answers to the preceding question. 5. Find the greater of two numbers, whose sum is 60, and product 800. 6. Find the less of two numbers, whose sum is 60 and product 800. These two questions, fifth and sixth, give the same equation, and that equation gives two values of x. One answers one of the questions, and the other, the other. 7. If the second power of a number be subtracted from the number, the remainder will be. What is the number? 8. If the second power of a number be subtracted from twelve times the number, the remainder will be 20. What is the number? It will be seen that each of the two preceding questions, has two answers, and the two values of x in the equations, are the two answers to the questions. 9. One number is 6 more than another, and the product of the two numbers is 216. What are the numbers? x = the less, x+6= the greater. Let x2+6x=216 x2+6x+9=225 When we supposed x the less number, we obtained x2 + 6 x 216. When we supposed x = the greater number, we found x2-6x=216. These two different equations give the same two values of x, except that they have different signs In the different equations; the value of x, which is plus in one, is minus in the other. Solve the following problems in the same manner: 10. One number is 4 more than another, and their product 192. What are the numbers? 11. A man bought two pieces of cloth which differed in length by 16 yards, for 160 dollars. Each piece cost as many dollars per yard as the other contained yards. How many yards were there in each piece? 12. Divide 20 into two parts, so that their product shall be 104. |