Question

One positive integer is 3 less than a second positive integer. The sum of the squares of the two integers is 65. Find both positive integers.

A) 4 and 7
B) 5 and 8
C) 6 and 9
D) 3 and 6

Answer 1

The first positive integer is 7 and the second positive integer is 10.

Step-by-step explanation:

Let's assume the first positive integer is x. Since the second positive integer is 3 more than x, it can be represented as x + 3.

According to the given information, the sum of the squares of these two integers is 65. So we can write the equation as: x² + (x + 3)² = 65.

Expanding the equation, we get: x² + x² + 6x + 9 = 65.

Combining like terms, we have: 2x² + 6x + 9 = 65.

Subtracting 65 from both sides, we get: 2x² + 6x - 56 = 0.

Factoring the quadratic equation, we have: (2x - 14)(x + 4) = 0.

Setting each factor equal to zero, we find two possible values for x: x = 7 and x = -4.

Since we are looking for positive integers, the first positive integer is 7 and the second positive integer is 7 + 3 = 10.