SOLUTION: The question is: Find two positive integers tha differ by 4 and whose product is 221. I know the answer is 13 and 17, but would like to know how to arrive at that algebraicly.

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Question 211955: The question is:
Find two positive integers tha differ by 4 and whose product is 221. I know the answer is 13 and 17, but would like to know how to arrive at that algebraicly.
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
The question is:

Find two positive integers that differ by 4 and whose product is 221. I know the answer is 13 and 17, but would like to know how to arrive at that algebraically.

Step 1. Two Positive Integers that differ by 4. Say n is one positive integer. Since the two numbers differ by 4, then the other must can be either n+4 or n-4. We'll choose n+4 but you can use n-4 and get similar results.

Step 2. Product is 221. This means that n(n+4)=221.


Step 3. Multiply the equation is step 2. n%28n%2B4%29=n%5E2%2B4n=221. This will reduce to a quadratic equation given as


n%5E2%2B4n-221=0

where we subtracted 221 from both sides of the equation in the first equation of Step 3.

Step 4. Now follow the process of solving a quadratic equation

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
where a=1, b=4 and c=-221

The steps are shown below. The solutions to the quadratic equation equation below are 13 and -17 and will intercept the x-axis in the parabola such that y=0. However, please ignore the graph for the moment since solution exceeded the limits for the graph.

Step 4a. Since we want a positive numbers then choose n=13. Therefore n+4=17.

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B4x%2B-221+=+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=%284%29%5E2-4%2A1%2A-221=900.

Discriminant d=900 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-4%2B-sqrt%28+900+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%284%29%2Bsqrt%28+900+%29%29%2F2%5C1+=+13
x%5B2%5D+=+%28-%284%29-sqrt%28+900+%29%29%2F2%5C1+=+-17

Quadratic expression 1x%5E2%2B4x%2B-221 can be factored:
1x%5E2%2B4x%2B-221+=+1%28x-13%29%2A%28x--17%29
Again, the answer is: 13, -17. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B4%2Ax%2B-221+%29



For step-by-step videos in Introduction to Algebra, please visit http://www.FreedomUniversity.Tv/courses/IntroAlgebra