variable stands for a number of men fractional solutions should be rejected. If only such results are obtained, the problem is self-contradictory. Often negative solutions should be rejected, as when the result indicates a negative number of digits in a number. Imaginary or complex (p. 72) results in general mean that the conditions of the problem cannot be realized. PROBLEMS 1. The product of and of a certain number is 500. What is the number? 2. There are two numbers one of which is less than 100 by as much as the other exceeds it. Their product is 9831. What are the numbers? 3. The sum of the square roots of two numbers is √74. One of the numbers is less than 37 by as much as the other exceeds it. What are the numbers? 4. A man sells oranges for as many cents apiece as he has oranges. He sells out for $3. How many had he? 5. If the perimeter of a square is 100 feet, how long is its diagonal? 6. A man sells goods and makes as much per cent as the number of dollars in the buying price. He made $245. What was the selling price? 7. Two bodies A and B move on the sides of a right angle. A is now 123 feet from the vertex and is moving away from it at the rate of 239 feet per second. B is 239 feet from the vertex and moves toward it at a rate of 123 feet per second. At what time (past or present) are they 850 feet apart ? 8. What is the number 29 times whose square exceeds itself by 190? 9. What numbers have a sum equal to 53 and a product equal to 612 ? 10. The sum of the squares of two numbers whose difference is 12 is found to be 1130. What are the numbers? 11. By what number must one increase each factor of 24. 20 so that the product shall be 540 greater? 12. What numbers have a quotient 4 and a product 900? 13. Two numbers are in the ratio of 4: 5. Increase each by 15 and the difference of their squares is 999. What are the numbers? 14. If 4 is divided by a certain number, the same result is obtained as if the number had been subtracted from 41. What is the number? 15. Separate 900 into two parts such that the sum of their reciprocal values is the reciprocal of 221. 16. The denominator of a fraction is greater by 4 than the numerator. Decrease the numerator by 3 and increase the denominator by the same, and the resulting fraction is half as great as the original one. What is the original fraction? 17. The numerator and denominator of a fraction are together equal to 100. Increase the numerator by 18 and decrease the denominator by 16, and the fraction is doubled. What is the fraction? 18. A number consists of two digits whose sum is 10. of the digits and multiply the resulting number by the the result is 2944. What is the number? Reverse the order original one, and 19. The sum of two numbers is 200. The square root of one increased by the other is 44. What are the numbers? 20. The difference of two numbers is 10, and the difference of their cubes is 20530. What are the numbers? 21. Around a rectangular flower bed which is 3 yards by 4 yards there extends a border of turf which is everywhere of equal breadth and whose area is ten times the area of the bed. How wide is it? 22. Two bicyclists travel toward each other, starting at the same time from places 51 miles apart. One goes at the rate of 9 miles an hour. The number of miles per hour gone by the other is greater by 5 than the number of hours before they meet. How far does each travel before they meet? 23. A printed page has 15 lines more than the average number of letters per line. If the number of lines is increased by 15, the number of letters per line must be decreased by 10 in order that the amount of matter on the two pages may be the same. How many letters are there on the page? 24. A merchant buys goods for a certain sum. The cost of handling them was 5% of their cost price. He sells for $504, gaining as much per cent as the cost price was in dollars. What was the cost price? 25. A man had $8000 at interest. He increased his capital by $100 at the end of each year, apart from his interest. At the beginning of the third year he had $8982.80. What per cent interest did his money draw? 26. Two men A and B can dig a trench in 20 days. It would take A 9 days longer to dig it alone than it would B. How long would it take B alone ? 27. A cistern is emptied by two pipes in 6 hours. How long would it take each pipe to do the work if the first can do it in 5 hours less time than the second? 28. A párty procures lunch at a restaurant for $15. If there had been 5 less in the party, each member would have paid 15 cents more without affecting the amount of the entire bill. How many were in the party? 29. A party pays $12 for accommodations. Had there been 4 less in the party, and if each person had paid 25 cents less, the bill would have been $15. How many were in the party? 30. A grocer sells his stock of butter for $15. If he had had 5 pounds less in stock, he would have been obliged to charge 10 cents more a pound to realize the same amount. How many pounds had he in stock? 31. A man buys lemons for $2. If he had received for that money 50 more lemons, they would have cost him 2 cents less apiece. What was the price of each lemon? 32. It took a number of men as many days to dig a ditch as there were men. If there had been 6 more men, the work would have been done in 8 days. How many men were there? 95. Theorems regarding quadratic equations. In this and the following sections we prove several theorems concerning quadratic equations. Similar theorems are later proved in general for equations of higher degree. THEOREM. If a is a root of the equation ax2 + bx + c = = 0, then xa is a factor of its left-hand member, and conversely. (1) The fact that a is a root of the equation is equivalent to the assertion of the truth of the identity aa2 + bx + c = 0, by definition of the root of an equation (§ 53). Since the remainder vanishes by hypothesis, ax2 + bx + c is exactly divisible by x α. Conversely, we have already seen (p. 3) that if x a is a factor of an equation, a is a root, since replacing x by a would make that factor vanish. 96. THEOREM. A quadratic equation has only two roots. Given the equation ax2 + bx + c = 0 with the roots a and ẞ, to prove that the equation has no other root, as y, distinct from a and ß. Since a and ẞ are roots of the equation, x α and x Thus our equation may be written in the form B are factors. (1) But in order that any product of numerical factors should vanish one of the factors must vanish. Thus either a = 0, or yα = 0, or yẞ = 0. But, by hypothesis, y a and yẞ, so the last two factors cannot vanish. Thus a = = 0. This would, however, reduce our equation to a linear equation, which is contrary to our hypothesis that the equation is quadratic. Thus the assumption that we have three distinct roots leads to a contradiction. COROLLARY I. If a quadratic equation is satisfied by more than two distinct values of the variable, then each of the coefficients vanishes identically. The above proof shows that the coefficient of x2 must vanish. In the same way it can be shown that b = c = 0. COROLLARY II. If two quadratic equations have the same value for more than two values of the variable, then their coefficients are identical. Let ax2 + bx + c = a'x2 + b'x + c' for more than two values of x. Transpose, and we obtain (a − a ́) x2 + (b − b′') x + c − c′ = 0. We have then a quadratic equation satisfied by more than two values of x. Thus by Corollary I each of its coefficients must vanish. Thus a' = a, b' = b, c = c. This theorem taken with § 95 is equivalent to the statement that a quadratic equation can be factored in one and only one way. Thus if we cannot find other numbers y and & distinct from a and ẞ such that ax2 + bx + c = a (x − y) (x — 8), for then the equation would have roots distinct from a and B. 97. THEOREM. If the equation x2 + bx + c = 0, where b and c are integers, has rational roots, those roots must be integers. (1) to be a rational fraction reduced to its lowest terms and Thus some factor of a must be contained in p (§ 69), which contradicts the hypothesis that the fraction Ρ is already reduced to its lowest terms. q 98. Nature of the roots of a quadratic equation. The equation These expressions afford an immediate arithmetic means of determining the nature of the roots of the given equation when a, b, and c have numerical values and a 0. In fact an inspection of the value of 2 4 ac is sufficient to determine the nature of the roots of (1). I. When b2-4 ac is negative, the roots are imaginary (§ 89). II. When b2-4 ac=0, the roots are real and equal. In this case x1 = x2 III. When b2 b 2 a 4 ac is positive, the roots are real and distinct. IV. When b2-4 ac is positive and a perfect square, the roots are real, distinct, and rational. |