Lesson 2 · from §57 of the 1910 text
Problems Solved by Factoring
Many problems lead to quadratic equations. Solve by factoring, then check which roots actually make sense: a length can't be negative.
Example. A triangle has area 24 square feet, and its altitude is 2 feet longer than its base. Let x be the base; the altitude is x + 2. The altitude of a triangle is the perpendicular distance from a corner to the opposite side (the base), and the area is half their product: \frac{x(x + 2)}{2} = 24 \;\Rightarrow\; x^2 + 2x - 48 = 0 \;\Rightarrow\; (x + 8)(x - 6) = 0. The root -8 makes no sense for a length, so the base is 6 feet and the altitude 8.
A trapezoid has two unequal parallel sides, its bases b and c. Its altitude a is the distance between them, and its area is \dfrac{a(b + c)}{2}.
Watch the units: a square yard is 9 square feet, and a square meter is 10,000 square centimeters.
Quick check
A rectangle has area 40 square feet and is 3 feet longer than it is wide. Find its dimensions.
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.
Solve each problem. When asked for every number that works, list them all.
The square of a number plus the number itself is 90. Find every number that works.
If twice a number is subtracted from its square, the remainder is 35. Find every number that works.
If twice a number plus 5 is added to its square, the result is 148. Find every number that works.
Four times the square of a number equals seven times the number. Find every number that works.
A number is added to 20, and the same number is added to 21. The product of the two sums is 930. Find every number that works.
A number is subtracted from 17, and also from 23. The product of the remainders is 216. Find every number that works.
A number is subtracted from 27 and added to 21. The product of the results is 540. Find every number that works.
If a number is added to 15 and subtracted from 22, the product of the sum and difference is 70 more than 23 times the number. Find every number that works.
The difference of two numbers is 6, and the difference of their squares is 120. Find the numbers.
If five times a number is taken from the square of three times the number, the result is eight times the square of the number. Find every number that works.
A lot of 2500 square feet is four times as deep as its frontage. Find its dimensions in feet.
A floor has an area of 24 square yards and is 6 feet longer than it is wide. Find its dimensions in feet.
A rectangular field of 216 square rods is 6 rods longer than it is wide. Find its dimensions in rods.
The sum of the squares of two consecutive positive numbers is 145. Find them.
The sum of the squares of two consecutive positive odd numbers is 290. Find them.
The sum of the squares of three consecutive positive odd numbers is 251. Find them.
An open (uncovered) square box 8 inches deep has 185 square inches of inside surface. Find the other inside dimensions.
A cube-shaped box has 294 square inches of outer surface. Find its edge.
A box is four times as long and three times as wide as it is deep, with 608 square feet of outer surface. Find its dimensions.
A box is 3 inches longer and 1 inch wider than it is deep, with 62 square inches of outer surface. Find its dimensions.
Triangles: area = \tfrac{ab}{2}, altitude a times base b, over 2.
A triangle has area 30 square feet and altitude 6 feet. Find the base.
A triangle's altitude is three times its base, and its area is 54 square feet. Find both.
A triangle's base is five times its altitude, and its area is 40 square feet. Find both.
A triangle has area 75 square meters, and its base is six times its altitude. Find both.
A triangle has area 24 square feet, and its altitude is 2 feet longer than its base. Find both.
A triangle's altitude is 3 feet longer than its base, and its area is 6 square yards. Find both in feet.
One leg of a right triangle is 2 feet longer than the other, and the area is 24 square feet. Find the legs.
A right triangle has area 30 square feet, and one leg is 7 feet longer than the other. Find the legs.
A triangle has area 2\tfrac58 square feet, and its base is 6 inches longer than twice its altitude. Find both in inches.
A triangle has area 4 square yards, and its altitude is 6 feet more than three times its base. Find both in feet.
A triangle has area 0.015 square meters, and its altitude is 5 centimeters shorter than its base. Find both in centimeters.
Trapezoids: area = \tfrac{a(b + c)}{2}, altitude a, parallel bases b and c.
Find the area of a trapezoid with bases 10 and 18 and altitude 12.
A trapezoid has altitude 8 inches and area 96 square inches, and one base is 4 inches longer than the other. Find each base.
One base of a trapezoid is 12 feet, the other is twice the altitude, and the area is 112 square feet. Find the altitude.
A trapezoid's altitude is \tfrac12 the shorter base, the shorter base is \tfrac23 the longer, and the area is 360 square feet. Find the bases and altitude.
One base of a trapezoid is 4 feet longer than the altitude, the other is 6 feet longer, and the area is 66 square feet. Find the bases and altitude.
The bases of a trapezoid are 8 and 12 feet longer than the altitude, and the area is 16 square yards. Find the bases and altitude in feet.
One base of a trapezoid is 4 feet longer than the other, the altitude is half the sum of the bases, and the area is 4 square yards. Find the bases and altitude in feet.
A trapezoid has area 10 square yards; the altitude equals one base, and the other base is 2 feet longer than the altitude. Find the bases and altitude in feet.
One base of a trapezoid exceeds the other by 10 feet, the altitude is 2 feet more than five times the shorter base, and the area is 22 square yards. Find the altitude and bases in feet.
A trapezoid has area 0.09 square meters; one base is twice the other, and the altitude is 10 centimeters less than the longer base. Find the bases and altitude in centimeters.