The product of two consecutive positive integers is 306. Form the quadratic equation to find x, if x denotes the smaller integer.

To solve the problem of finding two consecutive positive integers whose product is 306, we can start by defining the integers. Let's denote the smaller integer as \( x \). Consequently, the next consecutive integer would be \( x + 1 \). The relationship between these integers can be expressed mathematically as follows:

Setting Up the Equation

The product of the two integers can be represented by the equation:

x(x + 1) = 306

Expanding the Equation

Now, let's expand this equation:

x^2 + x = 306

Rearranging to Standard Form

To form a standard quadratic equation, we need to rearrange it so that one side equals zero:

x^2 + x - 306 = 0

Understanding the Quadratic Equation

This equation is now in the standard form of a quadratic equation, which is generally written as:

ax^2 + bx + c = 0

In our case, \( a = 1 \), \( b = 1 \), and \( c = -306 \).

Finding the Roots

To find the values of \( x \), we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

Substituting in our values:

x = (−1 ± √(1² - 4(1)(−306))) / 2(1)

This simplifies to:

x = (−1 ± √(1 + 1224)) / 2

x = (−1 ± √1225) / 2

Calculating the square root gives us:

√1225 = 35

Now substituting back into the equation:

x = (−1 ± 35) / 2

Calculating the Possible Values

This results in two potential solutions:

• x = (34) / 2 = 17
• x = (−36) / 2 = −18 (not a valid solution since we are looking for positive integers)

Identifying the Consecutive Integers

Since we are only interested in positive integers, we take \( x = 17 \). Therefore, the two consecutive integers are:

• 17
• 18

Verification

To ensure our solution is correct, we can check the product:

17 × 18 = 306

This confirms that our integers are indeed correct. Thus, the smaller integer \( x \) is:

17