Introduction to Quadratic Application Word Problems
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.
Solving for Consecutive Numbers
Examples
Example 1:
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

x ^{2} + x

= 240

x ^{2} + 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.)
Example 2:
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).
x ( x + 2)

= 528

x ^{2} + 2 x

= 528

x ^{2} + 2 x – 528

= 0

( x – 22)( x + 24)

= 0

The two solutions are 22 and 24, and –24 and –22.
Solving for NonConsecutive Numbers Example
Example
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.
Let x + 5 represent the other number

Let x − 5 represent the other number

x ( x + 5) = 104

x ( x − 5) = 104

x ^{2} + 5 x = 104

x ^{2} − 5 x = 104

x ^{2} + 5 x − 104 = 0 ( x + 13)( x − 8) = 0 x − 8 = 0 ( x + 13 = 0 leads to a +8 +8 negative solution) x = 8 The numbers are 8 and 8 + 5 = 13.

x ^{2}  5 x − 104 = 0 ( x  13)( x + 8) = 0 x − 13 = 0 ( x + 8 = 0 leads to a +13 +13 negative solution) x = 13 The numbers are 13 and 13 − 5 = 8.

Quadratic Applications Practice Problems
Practice
 The product of two consecutive odd numbers is 399. Find the numbers.
 The product of two consecutive numbers is 380. Find the numbers.
 The product of two consecutive numbers is 650. Find the numbers.
 The product of two consecutive even numbers is 288. What are the numbers?
 Two numbers differ by 7. Their product is 228. What are the numbers?
Solution

Let x = first number x + 2 = second number
There are two solutions: 19 and 21, and –21 and –19.

Let x = first number x + 1 = second number
x ( x + 1)

= 380

x ^{2} + x

= 380

x ^{2} + x – 380

= 0

( x + 20)( x – 19)

= 0

There are two solutions: 19 and 20, and –19 and –20.

Let x = first number x + 1 = second number
x ( x + 1)

= 650

x ^{2} + x

= 650

x ^{2} + x – 650

= 0

( x – 25)( x + 26)

= 0

There are two solutions: 25 and 26, and –25 and –26.

Let x = first number x + 2 = second number
x ( x + 2)

= 288

x ^{2} + 2 x

= 288

x ^{2} + 2 x – 288

= 0

( x – 16)( x + 18)

= 0

There are two solutions: 16 and 18, and –16 and –18.

Let x = first number x + 7 = second number
x ( x + 7)

= 228

x ^{2} + 7 x

= 228

x ^{2} + 7 x – 228

= 0

( x – 12)( x + 19)

= 0

There are two solutions: 12 and 19, –12 and –19.
Practice problems for this concept can be found at: Algebra Quadratic Applications Practice Test.
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From Algebra Demystified: A SelfTeaching Guide. Copyright © 2003 by The McGrawHill Companies, Inc. All Rights Reserved.