SOLUTION: Does anyone know how to go about this problem? The difference of two positive numbers is 3 and the sum of their squares is 65. Find the numbers.

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Question 214334: Does anyone know how to go about this problem? The difference of two positive numbers is 3 and the sum of their squares is 65. Find the numbers.
Answer by drj(1380) About Me  (Show Source):
You can put this solution on YOUR website!
Does anyone know how to go about this problem? The difference of two positive numbers is 3 and the sum of their squares is 65. Find the numbers.

Step 1. Let y be one positive and the larger number and x be the other positive and smaller number.

Step 2. The problem statement provides the following relationships or equations:

y-x=3 or y=x%2B3 (A)

x%5E2%2By%5E2=65 (B)

Step 3. Substitute (A) into (B)

x%5E2%2B%28x%2B3%29%5E2=65

x%5E2%2Bx%5E2%2B6x%2B9=65

Step 4. Subtract 65 from both sides of equation.

x%5E2%2Bx%5E2%2B6x%2B9-65=65-65

2x%5E2%2B6x-56=0

x%5E2%2B3x-28=0

Step 5. We can use the quadratic formula given as x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+

where a=1, b=3, and c=-28

Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B3x%2B-28+=+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=%283%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.

x%5B1%5D+=+%28-%283%29%2Bsqrt%28+121+%29%29%2F2%5C1+=+4
x%5B2%5D+=+%28-%283%29-sqrt%28+121+%29%29%2F2%5C1+=+-7

Quadratic expression 1x%5E2%2B3x%2B-28 can be factored:
1x%5E2%2B3x%2B-28+=+1%28x-4%29%2A%28x--7%29
Again, the answer is: 4, -7. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B3%2Ax%2B-28+%29



Step 6. Based on the above steps, select 4 since we want a positive number.

With x=4, then y=3+4=7.

Check if x%5E2%2By%5E2=65

4%5E2%2B7%5E2=65

16%2B49=65

65=65 which is a true statement.

Step 7. ANSWER: The numbers are 4 and 7.

I hope the above steps were helpful.

For free Step-By-Step Videos on Introduction to Algebra, please visit http://www.FreedomUniversity.TV/courses/IntroAlgebra or for Trigonometry visit http://www.FreedomUniversity.TV/courses/Trigonometry.

And good luck in your studies!

Respectfully,
Dr J