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

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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) About Me  (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
:
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the two positive integers are 6 and 11
**************************************
:

Answer by Alan3354(69443) About Me  (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.
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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."
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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 ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-17x%2B65+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-17%29%5E2-4%2A1%2A65=29.

Discriminant d=29 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--17%2B-sqrt%28+29+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-17%29%2Bsqrt%28+29+%29%29%2F2%5C1+=+11.1925824035673
x%5B2%5D+=+%28-%28-17%29-sqrt%28+29+%29%29%2F2%5C1+=+5.80741759643275

Quadratic expression 1x%5E2%2B-17x%2B65 can be factored:
1x%5E2%2B-17x%2B65+=+%28x-11.1925824035673%29%2A%28x-5.80741759643275%29
Again, the answer is: 11.1925824035673, 5.80741759643275. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-17%2Ax%2B65+%29

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j =~ 11.19 or 5.8

Answer by ikleyn(52790) About Me  (Show Source):
You can put this solution on YOUR website!
.
11 and 6.

Simple and easy mental exercise.

3 seconds.