Lesson 4 · from §107 of the 1910 text
Problems Leading to Quadratic Systems
Problems leading to quadratic systems
Many problems are stated most naturally with two unknowns. When the conditions lead to a system of the second degree, solve it by one of the methods of this chapter. Reject all results which do not satisfy the conditions of the problems: a negative length, a fraction of a person, and so on.
Example. The sum of two numbers is 10 and the sum of their squares is 58. Find the numbers.
Let g = the greater number and l = the less. Then
\begin{cases} g + l = 10 & (1) \\ g^2 + l^2 = 58 & (2) \end{cases}
Squaring (1), g^2 + 2gl + l^2 = 100; subtracting (2), 2gl = 42. Then (g - l)^2 = 58 - 42 = 16, so g - l = 4 (the greater minus the less is positive). With (1), g = 7 and l = 3.
Units. In the field problems, 1 acre = 160 square rods; in problem 11, 1 square foot = 144 square inches.
Quick check
The sum of two numbers is 10 and the sum of their squares is 58. Find the numbers.
From the history of algebra
Gerbert. In the tenth century there lived at least one man of remarkable mathematical talents, Gerbert by name, who wrote on both algebra and geometry. One of the most difficult problems that he set himself to solve was that of finding the legs of a right triangle whose area and hypotenuse were known, and he obtained the correct result. This is very remarkable when we consider the crude notation in which he did his work. One may compare the development of algebra at that time with that of the present time by observing that the problem, the solution of which was a brilliant achievement for one of the keenest mathematicians of the Middle Ages, is now solved with ease during the first year of algebra: it is problem 4 below.
Gerbert was prominent in the church, and in 999 was elected pope and assumed the title of Sylvester II.
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Reject all results which do not satisfy the conditions of the problems. Where two answers both satisfy a problem, give both.
Find two numbers whose difference is 3, and the difference of whose squares is 45.
The sum of two numbers is 18 and the sum of their squares is 170. Find the numbers.
Find two numbers whose product is 108 and whose quotient is \frac34.
The area of a right triangle is 84 square feet and its hypotenuse is 25 feet. Find the legs.
A rectangular field is 39 rods longer than it is wide, and its area is 10 acres. Find the length and the width.
The difference of the areas of two squares is 208 square feet and the difference of their perimeters is 32 feet. Find a side of each square.
The area of a rectangular field is 2\frac14 acres and one diagonal is 41 rods. Find the perimeter of the field.
The perimeter of a rectangle is 92 feet and its area is 504 square feet. Find the length and the width.
A mean proportional between two numbers is \sqrt{40}. The sum of their squares is 89. Find the numbers.
The value of a certain fraction is \frac23. The fraction is squared and 50 is subtracted from the numerator and from the denominator of this result. The value of the fraction thus formed is \frac27. Find the original fraction.
The base of a triangle is 5 inches longer than its altitude. Its area is 1\frac{1}{24} square feet. Find the base and the altitude of the triangle.
The volumes of two cubes differ by 1647 cubic inches. Their edges differ by 3 inches. Find the edge of each.
The sum of the radii of two circles is 27 inches and the difference of their areas is 81\pi square inches. Find the radii.
The perimeter of a rectangle is 3c and its area is \dfrac{c^2}{2}. Find its dimensions.
The area of a right triangle is 2a^2 - 2b^2 and its hypotenuse is 2\sqrt{2a^2 + 2b^2}. Find the legs.
The perimeter of a right triangle is 40 feet and its area is 60 square feet. Find the legs and the hypotenuse.
Do positive integers differing by 3 exist such that the sum of their squares is 115?
If a 2-digit number be multiplied by the sum of its digits, the product is 576. And if three times the sum of its digits be added to the number, the result is expressed by the digits in reverse order. Find the number.
The annual income from a certain investment is $48. If the principal were $200 more and the rate of interest 1\% less, the annual income would be $2 more. Find the principal and the rate.
A wheelman leaves A and travels north. At the same time a second wheelman leaves a point 3 miles east of A and travels east. An hour after starting the shortest distance between them is 17 miles, and 2\frac12 hours later the distance is 53 miles. Find the rate of each.