Quadratic Applications
Most of the problems in this chapter are not much different from the word problems in previous chapters. The only difference is that quadratic equations are used to solve them. Because quadratic equations usually have two solutions, some of these applied problems will have two solutions. Most will have only one-one of the ‘‘solutions’’ will be invalid. More often than not, the invalid solutions are easy to recognize.
Examples
The product
of two consecutive positive numbers is 240. Find the numbers.
Let x represent the first number. Because
the numbers are consecutive, the next number is one more than the first: x + 1
represents the next number. The product of these two numbers is x(x + 1), which
equals 240.
x(x + 1) = 240
x2 + x = 240
x2 + x – 240 = 0
(x – 15)(x + 16) = 0
x – 15 = 0 (x
+ 16) = 0 leads to a negative solution)
+15 +15
x = 15
The
consecutive positive numbers are 15 and 16.
(This problem could have been set up with x
representing the first number and x – 1 representing the second number.)
The product
of two consecutive even numbers is 528. What are the numbers?
Let x represent the first number.
Consecutive even numbers (and consecutive odd numbers) differ by two, so let x +
2 represent the second number. Their product is x(x + 2).
The two
solutions are 22 and 24, and –24
and –22.
Two positive numbers differ by five. Their
product is 104. Find the two numbers.
Let x represent the first number. If x
differs from the other number by five, then the other number could either be x +
5 or x – 5; it does not matter which representation you use. We will work this
problem with both representations.
Practice
1. The
product of two consecutive odd numbers is 399. Find the numbers.
2. The
product of two consecutive numbers is 380. Find the numbers.
3. The
product of two consecutive numbers is 650. Find the numbers.
4. The
product of two consecutive even numbers is 288. What are the numbers?
5. Two
numbers differ by 7. Their product is 228. What are the numbers?
Solution
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