Chapter 11 QuaDraTiC appliCaTionS 389

Two positive numbers differ by 5. Their product is 104. Find the two

numbers.

We let x represent the first number. If x differs from the other number by 5,

then the other number could either be x + 5 or x – 5; it does not matter

which representation we use. To see why it does not matter, 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 − 5x = 104

( + 13 = 0 leads

xx

xx

xx

(^251040)
13 80
80
+− =
+−=
−=
()()
to a negative
solution)
++

88
x 8
xx
xx
xx
(^251040)
13 80
13 0
−− =
−+=
−=
()()
( + 8 = 0 leads tto a
++ negative solution)

13 13
x 13
The numbers are 8 and 8 + 5 = 13. The numbers are 13 and 13 − 5 = 8.
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. Find the numbers.


5. Two numbers differ by 7. Their product is 228. Find the numbers.

✔SOLUTIONS

1. Let x = first number and x + 2 = second number.

(^) xx
xx
xx
xx
()
()()
+=
+=
+−=
−+=
2 399
2 399
2 399 0
19 21 0
2
2
x
x
x
−=

+=
19 0
19
221
x
x
x
+=
=−
+=−
21 0
21
219
There are two solutions: 19 and 21, and −21 and −19.
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.

PRACTICE

1. The product of two consecutive odd numbers is 399. Find the numbers.