The product of two consecutive positive integers is 306. We need to find the integers.

Let's denote the two consecutive integers as x and x+1.

Since the product of these two integers is 306, we can write the equation as:

x(x+1) = 306

Expanding this equation gives:

x^2 + x - 306 = 0

This is a quadratic equation in the form ax^2 + bx + c = 0, where a = 1, b = 1, and c = -306.

We can solve this equation by factoring, completing the square, or using the quadratic formula. In this case, factoring is the simplest method.

The factors of 306 that add up to 1 are -18 and 17, so we can write the equation as:

(x - 18)(x + 17) = 0

Setting each factor equal to zero gives the solutions x = 18 and x = -17.

Since the problem specifies that the integers are positive, the two consecutive integers are 18 and 19.