PROBLEMS PRODUCING PURE EQUATIONS ABOVE THE FIRST DEGREE. 172. Though the numerical negative values obtained in solving the following Problems satisfy the equations formed in accordance with the given conditions, they are practically inadmissible, and are therefore not given in the answers. 1. A gentleman being asked how many dollars he had in his purse, replied, "If you add 21 to the number and subtract 4 from the square root of the sum, the remainder will be 6." How many had he? 2. Divide 20 into two parts whose cubes shall be in the proportion of 27 to 8. 3. What two numbers are those less as 8: 3, and the sum of whose Ans. 12 and 8. whose sum is to the squares is 136? Ans. 10 and 6. 4. What number is that whose half multiplied by its third gives 54? 5. What number is that whose fourth and seventh multiplied together gives 463 ? 6. There is a rectangular field length is to its breadth as 8: 5. breadth? Ans. 36. containing 4 acres whose What is its length and 7. There are two numbers whose sum is 17, and the greater is to the greater divided by What are the numbers? less divided by the the less as 64: 81. Ans. 8 and 9. 8. The sum of the squares of two numbers is 65, and the difference of their squares 33. What are the numbers? 9. The sum of the squares of two quantities is a, and the difference of their squares b. What are the quantities? Ans. ± √1⁄2 (a + b) and ±√1 (a — b). 10. A gentleman sold two fields which together contained 240 acres. For each he received as many dollars an acre as there were acres in the field, and what he received for the larger was to what he received for the smaller as 49: 25. What are the contents of each? Ans. Larger, 140; smaller, 100 acres. 11. What are the two quantities whose product is a and quotient b? Ans. a ab and ± 12. What two numbers are as m: n, the sum of whose squares is a? Ans. ± m√ a and ± n√a √(m2 + n2) 13. What two numbers are as m:n, the difference of 14. Several gentlemen made an excursion, each taking $484. Each had as many servants as there were gentlemen, and the number of dollars which each had was four times the number of all the servants. How many gentlemen were there? Ans. 11. 15. Find three numbers such that the product of the first and second is 12; of the second and third, 20; and the sum of the squares of the first and third, 34. SECTION XIX. AFFECTED QUADRATIC EQUATIONS. 173. AN AFFECTED QUADRATIC EQUATION is one that contains both the first and second powers of the unknown quantity; as, 174. Every affected quadratic equation can be reduced to the form x2 + bx = c, in which b and c represent any quantities whatever, positive or negative, integral or fractional. For all the terms containing can be collected into one term whose coefficient we will represent by a; all the terms containing x can be collected into one term whose coefficient we will represent by d; and all the other terms can be united, whose aggregate we will represent by ẻ. Therefore every affected quadratic equation can be reduced to the form 175. The first member of the equation 2 + bx = c cannot be a perfect square. (Art. 145, Note 2.) But we know that the square of a binomial is the square of the first term plus or minus twice the product of the two terms plus the square of the last term; and if we can find the third term which will make x2 + bx a perfect square of a binomial, we can then reduce the equation x2 + bx = c. Since bx has in it as a factor the square root of x, x can be the first term of the square of a binomial, and bx the second term of the same square; and since the second term of the square is twice the product of the two terms of the binomial, the last term of the binomial must be the quotient arising from dividing the second term of the square by twice the square root of the first term of the ber, we have (2), an equation whose first member is a perfect square. Extracting the square root of each member of (2), and transposing, we obtain (4), or x = b 2 ± +c, which is a general expression for the value of x in any equation in the form of x2+ bx = C. Hence, as every affected quadratic equation can be reduced to the form x2+ bx = c, in which b and c represent any quantities whatever, positive or negative, integral or fractional, every affected quadratic equation can be reduced by the following RULE. Reduce the equation to the form x2 + bx = c, and add to each member the square of half the coefficient of x. Extract the square root of each member, and then reduce as in simple equations. |