SOLUTION: What is the smallest of 3 consecutive positive integers,if the product of the 2 smaller integers is 5 less than 5 times the largest integer?

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Question 955168: What is the smallest of 3 consecutive positive integers,if the product of the 2 smaller integers is 5 less than 5 times the largest integer?
Answer by macston(5194) About Me  (Show Source):
You can put this solution on YOUR website!
Smallest integer=n; middle integer=n+1; largest integer=n+2
(n)(n+1)=5(n+2)-5
n%5E2%2Bn=5n%2B10-5
n%5E2%2Bn=5n%2B5 Subtract (5n+5) from each side.
n%5E2-4n-5=0
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation an%5E2%2Bbn%2Bc=0 (in our case 1n%5E2%2B-4n%2B-5+=+0) has the following solutons:

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

Discriminant d=36 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--4%2B-sqrt%28+36+%29%29%2F2%5Ca.

n%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+36+%29%29%2F2%5C1+=+5
n%5B2%5D+=+%28-%28-4%29-sqrt%28+36+%29%29%2F2%5C1+=+-1

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

n=5 ANSWER: The smallest integer is 5.
CHECK:
middle integer=n+1=5+1=6
largest integer=n+2=5+2=7
Product of two smaller is 5 less than 5 times larger:
5(6)=5(7)-5
30=35-5
30=30