The product of two consecutive positive number is 182, find the number.

Answer

To solve the problem of finding two consecutive positive numbers whose product is 182, we can follow these steps:

### Step 1: Define the Variables
Let the first consecutive positive number be \( x \). Then, the next consecutive number will be \( x + 1 \).

### Step 2: Set Up the Equation
According to the problem, the product of these two numbers is 182. Therefore, we can write the equation:
\[ x(x + 1) = 182 \]

### Step 3: Expand the Equation
Expanding the left side gives us:
\[ x^2 + x = 182 \]

### Step 4: Rearrange the Equation
To form a standard quadratic equation, we rearrange it:
\[ x^2 + x - 182 = 0 \]

### Step 5: Apply the Quadratic Formula
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
For our equation \( x^2 + x - 182 = 0 \), we identify \( a = 1 \), \( b = 1 \), and \( c = -182 \).

### Step 6: Calculate the Discriminant
First, we calculate the discriminant \( b^2 - 4ac \):
\[ b^2 - 4ac = 1^2 - 4 \cdot 1 \cdot (-182) = 1 + 728 = 729 \]

### Step 7: Substitute into the Quadratic Formula
Now substituting the values into the quadratic formula:
\[ x = \frac{-1 \pm \sqrt{729}}{2 \cdot 1} \]
Since \( \sqrt{729} = 27 \):
\[ x = \frac{-1 \pm 27}{2} \]

### Step 8: Solve for \( x \)
Calculating the two possible values for \( x \):
1. \( x = \frac{-1 + 27}{2} = \frac{26}{2} = 13 \)
2. \( x = \frac{-1 - 27}{2} = \frac{-28}{2} = -14 \)

### Step 9: Identify the Positive Solution
Since we are looking for positive numbers, we take \( x = 13 \).

### Step 10: Find the Consecutive Number
The second consecutive number is:
\[ x + 1 = 13 + 1 = 14 \]

### Final Answer
The two consecutive positive numbers are **13 and 14**.

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