The product of two consecutive integer is 5 three times the larger integer, what are the integers?

To solve this problem, let's break it down step by step:

Let's assume the smaller integer is x.

According to the problem statement, the larger integer will be the next consecutive integer, which is x + 1.

The product of two consecutive integers is 5 three times the larger integer, which can be written as:

x(x + 1) = 5 * 3(x + 1)

Now, let's solve this equation:

x(x + 1) = 15(x + 1)

Expanding both sides of the equation:

x^2 + x = 15x + 15

Rearranging the equation:

x^2 + x - 15x - 15 = 0

Combining like terms:

x^2 - 14x - 15 = 0

Now, we can solve this quadratic equation either by factoring, completing the square, or using the quadratic formula. Since the quadratic equation can't be easily factored, let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

In this case, a = 1, b = -14, and c = -15:

x = (-(-14) ± √((-14)^2 - 4 * 1 * -15)) / (2 * 1)

Simplifying the equation:

x = (14 ± √(196 + 60)) / 2

x = (14 ± √256) / 2

x = (14 ± 16) / 2

Now, we have two possible values for x:

1. x = (14 + 16) / 2 = 30 / 2 = 15
2. x = (14 - 16) / 2 = -2 / 2 = -1

Therefore, the two consecutive integers are either 15 and 16 or -1 and 0.