SOLUTION: The product of two consecutive odd integers is 1 less than four times their sum. Find the two integers.

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Question 211832: The product of two consecutive odd integers is 1 less than four times their sum.
Find the two integers.
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
The product of two consecutive odd integers is 1 less than four times their sum.
Find the two integers.

Step 1. Let n be one odd integer.

Step 2. Let n+2 be the next consecutive odd integer.

Step 3. Translate problem statement into an equation such than n(n+2)=4(n+n+2)-1 since product is 1 less than four times their sum

where n(n+2) is the product of the two consecutive integers and (n+n+2) is the sum of the two integers

n%28n%2B2%29=4%28n%2Bn%2B2%29-1

n%5E2%2B2n=4%282n%2B2%29-1

n%5E2%2B2n=8n%2B8-1

n%5E2%2B2n=8n%2B7

Step 4. Simplify to obtain a quadratic equation. Subtract 8n+7 from both sides of equation to get the right side of equation to be zero.

n%5E2%2B2n-8n-7=8n%2B7-8n-7

n%5E2-6n-7=0

Step 5. Now we have a quadratic equation. So use quadratic formula given as

n+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where

a=1, b=-6 and c=-7

The following steps solves the quadratic equation

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation an%5E2%2Bbn%2Bc=0 (in our case 1n%5E2%2B-6n%2B-7+=+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=%28-6%29%5E2-4%2A1%2A-7=64.

Discriminant d=64 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--6%2B-sqrt%28+64+%29%29%2F2%5Ca.

n%5B1%5D+=+%28-%28-6%29%2Bsqrt%28+64+%29%29%2F2%5C1+=+7
n%5B2%5D+=+%28-%28-6%29-sqrt%28+64+%29%29%2F2%5C1+=+-1

Quadratic expression 1n%5E2%2B-6n%2B-7 can be factored:
1n%5E2%2B-6n%2B-7+=+1%28n-7%29%2A%28n--1%29
Again, the answer is: 7, -1. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-6%2Ax%2B-7+%29



Step 6. ANSWER: Based on the above steps, choose 7 since it is positive. The integers are then 7 and 9.

Step 7. Verify the answer using n(n+2)=4(n+n+2)+1 in Step 3.

%287%29%289%29=4%287%2B9%29-1

63=4%2816%29-1

63=63 So it checks out.

I hope the above steps were helpful.

And good luck in your studies!

Respectfully,
Dr J

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