# 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.
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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:

or (A)

(B)

Step 3. Substitute (A) into (B)

Step 4. Subtract 65 from both sides of equation.

Step 5. We can use the quadratic formula given as

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

 Solved by pluggable solver: SOLVE quadratic equation with variable Quadratic equation (in our case ) has the following solutons: For these solutions to exist, the discriminant should not be a negative number. First, we need to compute the discriminant : . Discriminant d=121 is greater than zero. That means that there are two solutions: . Quadratic expression can be factored: Again, the answer is: 4, -7. Here's your graph:

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

which is a true statement.

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

I hope the above steps were helpful.