Three consecutive positive integers are such that the sum of the square of the first and the product of other two is 46, fond the integers.

Answer

To solve the problem of finding three consecutive positive integers such that the sum of the square of the first integer and the product of the other two is 46, we can follow these steps:

### Step 1: Define the integers
Let the three consecutive integers be:
- First integer: \( x \)
- Second integer: \( x + 1 \)
- Third integer: \( x + 2 \)

### Step 2: Set up the equation
According to the problem, we need to express the condition given:
The sum of the square of the first integer and the product of the other two is equal to 46. This can be written as:
\[ x^2 + (x + 1)(x + 2) = 46 \]

### Step 3: Expand the equation
Now, we expand the product:
\[ (x + 1)(x + 2) = x^2 + 3x + 2 \]
Substituting this back into the equation gives:
\[ x^2 + x^2 + 3x + 2 = 46 \]

### Step 4: Combine like terms
Combining the terms results in:
\[ 2x^2 + 3x + 2 = 46 \]

### Step 5: Rearrange the equation
Now, we rearrange the equation to set it to zero:
\[ 2x^2 + 3x + 2 - 46 = 0 \]
\[ 2x^2 + 3x - 44 = 0 \]

### Step 6: Apply the quadratic formula
We will use the quadratic formula to solve for \( x \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 2 \), \( b = 3 \), and \( c = -44 \).

### Step 7: Calculate the discriminant
Calculate the discriminant:
\[ b^2 - 4ac = 3^2 - 4 \cdot 2 \cdot (-44) = 9 + 352 = 361 \]

### Step 8: Substitute back into the formula
Now, substituting back into the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{361}}{2 \cdot 2} \]
Since \( \sqrt{361} = 19 \), we have:
\[ x = \frac{-3 \pm 19}{4} \]

### Step 9: Calculate the possible values of \( x \)
Calculating the two possible values:
1. \( x = \frac{-3 + 19}{4} = \frac{16}{4} = 4 \)
2. \( x = \frac{-3 - 19}{4} = \frac{-22}{4} = -5.5 \) (not a positive integer)

Since we are looking for positive integers, we take \( x = 4 \).

### Step 10: Find the consecutive integers
Now, substituting back to find the integers:
- First integer: \( x = 4 \)
- Second integer: \( x + 1 = 5 \)
- Third integer: \( x + 2 = 6 \)

Thus, the three consecutive positive integers are:
\[ \boxed{4, 5, 6} \]