Lesson 5 · from §102 of the 1910 text
Problems Leading to Quadratics
Problems that lead to quadratic equations
Represent the unknown number by a letter, state the equation, and solve it by factoring, by completing the square, or by formula. A quadratic gives two roots, and both must be tested against the conditions of the problem. Reject all answers which do not satisfy the conditions of the problem: a negative width, a fractional number of persons, a time that has not happened.
Example. The product of two consecutive numbers is 132. Find the numbers.
Let n be the less number; then n + 1 is the greater, and
n(n + 1) = 132, \qquad n^2 + n - 132 = 0, \qquad (n - 11)(n + 12) = 0.
So n = 11 or -12: the numbers are 11 and 12, or -12 and -11. Both pairs satisfy the problem. Where a problem has two sets of answers, give both (for example, the less number 11, -12).
When the answer is a length, a rate, or a time, only the positive root has a meaning.
Try one:
Quick check
The product of two consecutive even numbers is 168. Find the less number (give both answers).
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 answers which do not satisfy the conditions of the problems. Where two answers both satisfy a problem, give both.
The sum of the square of a certain number and twice the number itself is 15. Find the number.
Find two numbers whose difference is 11 and whose product is 42.
If from twice the square of a certain number the number itself be taken away, the remainder is 45. Find the number.
Find two consecutive numbers whose product is 462.
Find two consecutive odd numbers whose product is 255.
Find three consecutive even numbers whose sum is \frac16 of the product of the first two.
A rectangular field is 16 rods longer than it is wide. Its area is 32 acres (1 acre = 160 square rods). Find the dimensions of the field.
The sum of a certain number and its reciprocal is 2\tfrac{1}{20}. Find the number.
The area of a triangular field is 5\tfrac58 acres. The base is 51 rods longer than the altitude. Find the base and the altitude.
Two square fields together contain 62.5 acres. A side of one is 20 rods longer than a side of the other. Find the side of each.
The area of a rectangle is 18 square inches less than twice the area of a square. The rectangle is 7 inches longer than the square, and a side of the latter equals the breadth of the rectangle. Find the side of the square.
The hypotenuse of a right triangle is 41 feet. One leg is 31 feet shorter than the other. Find the legs.
The legs of a right triangle are in the ratio of 3 : 4. The hypotenuse is 20. Find the legs.
One angle of a right triangle is 30 degrees and its longer leg is 9. Find, correct to two decimal places, the other two sides.
The hypotenuse of a right triangle is 10 and one leg is 5\sqrt3. Show that one angle of the triangle is 30 degrees and find the number of degrees in each angle of the triangle.
The area of a square in square feet and its perimeter in linear feet are expressed by the same number. Find the side.
The area of a square in square feet and its perimeter in inches are expressed by the same number. Find the side.
The area of a square in square inches and its perimeter in feet are expressed by the same number. Find the side of the square.