SOLUTION: given consecutive positive odd integers, the square of the smaller number minus 5 times the larger is 26. find the numbers

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Question 953551: given consecutive positive odd integers, the square of the smaller number minus 5 times the larger is 26. find the numbers
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
S=smaller; L=larger=S+2
S%5E2-5L=26 Substitute for L
S%5E2-5%28S%2B2%29=26 Subtract 26 from each side.
S%5E2-5S-10-26=0
S%5E2-5S-36=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation aS%5E2%2BbS%2Bc=0 (in our case 1S%5E2%2B-5S%2B-36+=+0) has the following solutons:

S%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=%28-5%29%5E2-4%2A1%2A-36=169.

Discriminant d=169 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--5%2B-sqrt%28+169+%29%29%2F2%5Ca.

S%5B1%5D+=+%28-%28-5%29%2Bsqrt%28+169+%29%29%2F2%5C1+=+9
S%5B2%5D+=+%28-%28-5%29-sqrt%28+169+%29%29%2F2%5C1+=+-4

Quadratic expression 1S%5E2%2B-5S%2B-36 can be factored:
1S%5E2%2B-5S%2B-36+=+1%28S-9%29%2A%28S--4%29
Again, the answer is: 9, -4. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-5%2Ax%2B-36+%29

S=9 ANSWER 1: The smaller number is 9.
L=S%2B2=11 ANSWER 2: The larger number is 11.
CHECK:
S%5E2-5L=26
9%5E2-5%2811%29=26
81-55=26
26=26