Find two consecutive positive integers, sum of whose squares are 365.

Options

A. 13,14
B. 12,13
C. 14,15
D. none of these

Explanation

To find two consecutive positive integers whose sum of squares is 365, we can follow these steps:

### Step 1: Define the Variables
Let the first consecutive positive integer be \( x \). Therefore, the second consecutive positive integer will be \( x + 1 \).

### Step 2: Set Up the Equation
According to the problem, the sum of the squares of these two integers is 365. We can express this mathematically as:
\[ x^2 + (x + 1)^2 = 365 \]

### Step 3: Expand the Equation
Now, we will expand the equation:
\[ x^2 + (x^2 + 2x + 1) = 365 \]
This simplifies to:
\[ 2x^2 + 2x + 1 = 365 \]

### Step 4: Rearrange the Equation
Next, we will rearrange the equation to set it to zero:
\[ 2x^2 + 2x + 1 - 365 = 0 \]
This simplifies to:
\[ 2x^2 + 2x - 364 = 0 \]

### Step 5: Divide the Equation
To simplify the equation, we can divide all terms by 2:
\[ x^2 + x - 182 = 0 \]

### Step 6: Factor the Quadratic Equation
Now we need to factor the quadratic equation \( x^2 + x - 182 = 0 \). We look for two numbers that multiply to \(-182\) and add to \(1\). The factors are \(14\) and \(-13\):
\[ (x + 14)(x - 13) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x + 14 = 0 \) → \( x = -14 \) (not a positive integer)
2. \( x - 13 = 0 \) → \( x = 13 \)

### Step 8: Identify the Consecutive Integers
Since we are looking for positive integers, we discard \( x = -14 \). Thus, we have:
- First integer: \( x = 13 \)
- Second integer: \( x + 1 = 14 \)

### Conclusion
The two consecutive positive integers are \( 13 \) and \( 14 \).