Let’s denote two consecutive positive integers as x and x + 1.
The product of these integers can be expressed as:
Product = x * (x + 1) = x2 + x
The sum of these integers is:
Sum = x + (x + 1) = 2x + 1
According to the problem, the product of the two integers is 55 more than their sum:
x2 + x = 2x + 1 + 55
Simplifying this equation:
x2 + x = 2x + 56
x2 + x – 2x – 56 = 0
x2 – x – 56 = 0
Next, we’ll factor the quadratic equation:
We need two numbers that multiply to -56 and add to -1. The numbers that satisfy this are -8 and 7:
(x – 8)(x + 7) = 0
This gives us two possible solutions:
x – 8 = 0 → x = 8
x + 7 = 0 → x = -7
Since we are looking for positive integers, we take x = 8.
Thus, the two consecutive positive integers are:
8 and 9.
To verify:
Product = 8 * 9 = 72
Sum = 8 + 9 = 17
72 = 17 + 55
Since this holds true, our solution is confirmed.
Therefore, the two consecutive positive integers are 8 and 9.