Algebra Demystified 2nd Ed

388 algebra De mystif ieD

Number Sense Problems

Most of the problems in this chapter aren’t much different from the word

problems in Chapter 8. The only difference is that we use quadratic equations

to solve them. Because most quadratic equations have two solutions, some of

the applied problems also have two solutions, too. Many, though, have one solu-

tion, and we have to decide which one of them is the correct solution.

EXAMPLES

The product of two consecutive positive numbers is 240. Find the numbers.

We 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 num-

ber. The product of these two numbers is x(x + 1), which equals 240.

xx

xx

xx

xx

x

()

()()

+=

+=

+− =

−+=

−=

1 240

240

240 0

15 16 0

15

2

2

001 (x+= 60 leadstoanegativesolution)

++

=

15 15

x 15

The consecutive positive numbers are 15 and 15 + 1 =16.

The product of two consecutive even numbers is 528. What are the

numbers?

We let x represent the first number. Consecutive even numbers (and con-

secutive odd numbers) differ by two, so we let x + 2 represent the second

number. Their product is x(x + 2).

xx

xx

xx

xx

()

()()

+=

+=

+−=

−+=

2 528

2 528

2 528 0

22 24 0

2

2

x

x

−=

++

=

22 0

22 22

22

x

x

+=

−−

=−

24 0

24 24

24

The two solutions are 22 and 24, and –24 and –22.

EXAMPLES

The product of two consecutive positive numbers is 240. Find the numbers.

We let

EXAMPLES

The product of two consecutive positive numbers is 240. Find the numbers.