SOLUTION: One integer is 3 less than another. The sum of their squares is 65. find the integers.

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Question 570838: One integer is 3 less than another. The sum of their squares is 65. find the integers.
Answer by JBarnum(2146) About Me  (Show Source):
You can put this solution on YOUR website!
a=b-3
a%5E2%2Bb%5E2=65
.
%28b-3%29%5E2%2Bb%5E2=65
b%5E2-6b%2B9%2Bb%5E2=65
2b%5E2-6b%2B9=65
2b%5E2-6b-56=0
2%28b%5E2-3b-28%29=0
find two numbers that multiply to get -28 and add to get -3
-7 and 4
2%28b-7%29%28b%2B4%29=0
b=7,-4
plug the b's into the first equation
a=4,-7
if you cant understand the quick solve above there's always the long quadratic solution below
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ab%5E2%2Bbb%2Bc=0 (in our case 1b%5E2%2B-3b%2B-28+=+0) has the following solutons:

b%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=%28-3%29%5E2-4%2A1%2A-28=121.

Discriminant d=121 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--3%2B-sqrt%28+121+%29%29%2F2%5Ca.

b%5B1%5D+=+%28-%28-3%29%2Bsqrt%28+121+%29%29%2F2%5C1+=+7
b%5B2%5D+=+%28-%28-3%29-sqrt%28+121+%29%29%2F2%5C1+=+-4

Quadratic expression 1b%5E2%2B-3b%2B-28 can be factored:
1b%5E2%2B-3b%2B-28+=+1%28b-7%29%2A%28b--4%29
Again, the answer is: 7, -4. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-3%2Ax%2B-28+%29