Word Problems

Problem: The product of two consecutive positive integers is $306$. Find them.

Solution: Let the integers be $x$ and $x + 1$.

$$x(x + 1) = 306$$ $$x^2 + x - 306 = 0$$

Using the quadratic formula:

$$x = \frac{-1 \pm \sqrt{1 + 1224}}{2} = \frac{-1 \pm \sqrt{1225}}{2} = \frac{-1 \pm 35}{2}$$ $$x = \frac{34}{2} = 17 \quad (\text{taking the positive root})$$

The two consecutive integers are $\boxed{17}$ and $\boxed{18}$.