Product of two consecutive positive integers is 306. Find them.

Step 1: Define variables

We are looking for two consecutive positive integers. We can represent the first integer as $x$ and the next consecutive integer as $x+1$.

Step 2: Formulate the equation

The problem states that the product of these two consecutive integers is 306. Therefore, we multiply $x$ by $(x+1)$ and set the product equal to 306.

Step 3: Expand and rearrange the equation

Expand the left side of the equation by distributing $x$ to both terms inside the parenthesis. Then, subtract 306 from both sides to set the quadratic equation to zero, which is the standard form $ax^2 + bx + c = 0$.

Step 4: Solve the quadratic equation by factorization

We need to find two numbers that multiply to give $-306$ and add up to $1$ (the coefficient of $x$). These numbers are $18$ and $-17$. We then split the middle term and factor by grouping to find the possible values for $x$.

Step 5: Find the possible values for x

By the zero product property, if the product of two factors is zero, then at least one of the factors must be zero. This gives us two possible values for $x$.

Step 6: Select the correct value for x

The problem specifies that the integers are positive. Therefore, we choose the positive value for $x$, which is 17. The value $x = -18$ is discarded because it is not a positive integer.

Step 7: Find the two consecutive integers

Substitute the chosen value of $x=17$ back into our definitions for the two integers. The first integer is 17, and the second integer is $17+1=18$.

Step 8: Verify the solution

To verify our answer, we multiply the two integers we found. Since their product is 306, our solution is correct.

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