SOLUTION: two positive integers have a sum 0f 17 and a product of 66. What are the integers?

Algebra.Com
Question 1132277: two positive integers have a sum 0f 17 and a product of 66. What are the integers?
Found 3 solutions by rothauserc, Alan3354, ikleyn:
Answer by rothauserc(4718)   (Show Source): You can put this solution on YOUR website!
1) i +j = 17
:
2) ij = 66
:
solve equation 1 for i
:
i = 17 -j
:
substitute for i in equation 2
:
(17 -j)j = 66
:
17j -j^2 = 66
:
j^2 -17j +66 = 0
:
(j-6)(j-11) = 0
:
j = 6 or j = 11
:
use equation 1, then if j=6 then i = 17 -6 = 11 and if j=11 then i = 17 -11 = 6
:
**************************************
the two positive integers are 6 and 11
**************************************
:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
Try pairs of factors of 66.
Should take less than 10 seconds to find them.
===================
Copied from the other tutor's solution:
------------
------------
1) i +j = 17
:
2) ij = 66
:
solve equation 1 for i
:
i = 17 -j
:
substitute for i in equation 2
:
(17 -j)j = 66
:
17j -j^2 = 66
:
j^2 -17j +66 = 0
At this point, you find a pair of factors of 66 with a sum of 17. Similar to "going around the barn to get to the door."
--------
But, if they're not integers, eg, the sum is 17 and the product is 65:
---
1) i +j = 17
:
2) ij = 65
:
i = 17 -j
:
j^2 -17j +65 = 0
:
(17 -j)j = 65
:
17j -j^2 = 65
j^2 - 17j + 65 = 0
Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)
Quadratic equation (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=29 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: 11.1925824035673, 5.80741759643275. Here's your graph:

-----
j =~ 11.19 or 5.8
Answer by ikleyn(52792)   (Show Source): You can put this solution on YOUR website!
.
11 and 6.

Simple and easy mental exercise.

3 seconds.


RELATED QUESTIONS

two positive integers have a sum of 21 and a product of 108. what are the... (answered by Sunny Day)
two positive integers have a sum of 14 and a product of 24. what are the... (answered by lynnlo,Alan3354)
two consecutive positive integers have a product of 240.find the integers. the... (answered by ikleyn)
two positive integers have a sum of 20 and a product of 96. what is one of the... (answered by Alan3354)
Two negative integers have a sum of -12 and a product of 11. What are the integers? (answered by ankor@dixie-net.com)
two negative integers have a sum of -13 and a product of 42. what are the... (answered by Cromlix)
two positive integers have a sum of 20 and a product of 91. what is the smaller... (answered by sarah_adam)
What two positive integers have the product of 5 and the sum of 8? (answered by stanbon)
Two integers have a difference of -11 and a sum of -3. What are the... (answered by Paul)