EXAMPLES. 1 55 ; 1. Given xo + 43 = 60; to find x. First, by completing the square, xo + 4x + 4 = 64 ; Then, transpos, 2, gives I = 6 or 10, the two roots., 2. Given a 6x + 10 = 65 ; to find x. First trans. 10 gives a ? 62 Then trans. 3, gives x = 11 or 5. First by transpos. 20, it is 2x2 + 8x = 90; And transp 2, gives x = 5 or 9. x + 3 27; And transp. 1, gives x = - or } 3. Given tr3 - 4x + 301 52 ; to find x, And transp. , gives x = = 7 or 61 $. Given ar? bx sc; to find x. 6 First by diy. by a, it is x? a 6 6 € 69 Then compl. the sq. gives - 2+ +-; 422 42 6 4ac + 6? And extrac. the root, gives x 3 4ac + 6? 6 + 2a 42% 2a 4, Given ** 2ax2 = 0; to find x. First by compl. the sq. gives **- 20x2 + 2 = ' +6; And And extract. the root, gives x2 -a= I Va? + b; And extract. the root, gives x = + vat v a2 + b. And thus, by always using similar words at each line, the pupil will resolve the following examples. EXAMPLES FOR PRACTICE. 5.7 l. Given x2 60 7 = 33; to find x, Ans. ¢ = 10 2. Given ? 10 = 14 : to find x. Ans, = 8. .3. Given 5.x2 + 4x - 90 = 114 ; to find x. Ans. x = 6. 4. Given jx2 - + 2 : 9; to find x. Ans. X = 4. 5. Given 3. * - 2x2 = 40; to find x. Ans x=2. 6. Given fot -- IV x = l; to find . Ans. x = 9. 7. Given jx2 +, for ; to find x. Ans. x = •727766. 8. Given x6 + 4x3 = 12; to find x. Ans. Is 2=1.259921. 9. Given x2 + 40 = a + 2; to find x. Ans.x = V2 + 6-2. QUESTIONS PRODUCING QUADRATIC EQUATIONS. Y = 2, 1. To find two numbers whose difference is 2, and pro duct 80. Then the first condition gives x - 8; These questions, like those in simple equations, are also solved by using as many unknown letters, as are the numbers required, for the better exercise in reducing equations ; not aiming at the shortest modes of solution, which would not afford so much useful practice. 2. To divide the number 14 into two such parts, that their product may be 48. gives ya 14y = - Y; 48 ; their squares 3. Given the sum of two numbers = 9, and the sum of = 45; to find those numbers. 92 + 36; 18; 4. What two numbers are those. whose sum, product, and difference of their squares, are all equal to each other? Let x and y denote the two numbers. y%. And therefore x = y +1=5+ tracting the root of 5, &c. they give x = 2:6180 +, and y = 16180 +. -5. There are four numbers in arithmetical progression, of which Vhich the product of the two extremes is 22, and that of the means 40 ; what are the numbers ? Let I = the less extreme, and y = the common difference ; - 22, Then extracing the root gives r+=; Hence the four numbers are 2, 5, 8, 11, Let x, y, and z, denote the three numbers sought. 21= 49 – 14y; other numbers ; QUESTIONS FOR PRACTICE od 1. WHAT number is that which added to its square makes Ans. 6. 2. To find two numbers such, that the less may be to the greater as the greater is to 12, and that the sum of their squares may be 45. Ans. 3 and 6. 3. What two numbers are those, whose difference is 2, and the difference of their cubes 98 ? Ans. 3 and 5. 4. What two numbers are those whose sum is 6, and the sum of their cubes 72? Ans. 2 and 4. 5 What two numbers are those, whose product is 20, and the difference of their cubes 61? Ans 4 and 5. 6. To divide the number 11 into two such parts, that the product of their squares may be 784. Ans. 4 and 7. 7. To divide the number 5 into two such parts, that the sum of their alternate quotients may be 45, that is of the two quotients of each part divided by the other. Ans. I and 4. 8. To divide 12 into two such parts, that their product may be equal to 8 times their difference. Ans. 4 and 8. 9. To divide the number 10 into two such parts, that the square of 4 times the less part, may be 112 more than the square of 2 times the greater. Ans. 4 and 6. 10 To find iwo numbers such, that the sum of their squares may be 89, and their sum multiplied by the greater may produce 104. Ans. 5 and 8. il. What number is that, which being divided by the product of its two digits, the quotient is 5; but when 9 is subtracted from it, there remains a number having the same digits inverted ? Ans. 32 12. To divide 20 into three parts, such that the continual product of all three may be 270, and that the difference of the first and second may be 2 less than the difference of the second and third. Ans. 5, 6, 9. 13. To find three numbers in arithmetical progression, such that the sum of their squares may be 56, and the sum arising by adding together 3 times the first and 2 times the second and 3 times the third, may amount to 28. Ans. 2, 4, 6. 14. To divide the number 13 into three such parts, that their squares may have equal differences, and that the sum of those squares may be 75. Ans. 1, 5, 7. 15. To find three numbers having equal differences, so that their sum may be 12, and the sum of their fourth powers 962. Ans. 3, 4, 5. 16. To find three numbers having equal differences, and such that the square of the least added to the product of the two greater may make 28, but the square of the greatest added co the product of the two less may make 44. Ans. 2, 4, 6. 17. Three |