The sum of a natural number and its positive square root is 132. Find the number.

Explanation

To solve the problem, we will follow these steps:

### Step 1: Define the variable
Let the natural number be \( x^2 \), where \( x \) is the positive square root of the number.

### Step 2: Set up the equation
According to the problem, the sum of the natural number and its positive square root is 132. This can be expressed as:
\[ x^2 + x = 132 \]

### Step 3: Rearrange the equation
Rearranging the equation gives us:
\[ x^2 + x - 132 = 0 \]

### Step 4: Factor the quadratic equation
Now, we need to factor the quadratic equation \( x^2 + x - 132 = 0 \). We look for two numbers that multiply to \(-132\) and add up to \(1\). The numbers \(12\) and \(-11\) satisfy this condition. Therefore, we can factor the equation as:
\[ (x + 12)(x - 11) = 0 \]

### Step 5: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x + 12 = 0 \) → \( x = -12 \)
2. \( x - 11 = 0 \) → \( x = 11 \)

### Step 6: Determine the valid solution
Since \( x \) represents the positive square root, we discard \( x = -12 \) and take \( x = 11 \).

### Step 7: Find the natural number
Now, we calculate the natural number:
\[ \text{Natural number} = x^2 = 11^2 = 121 \]

### Final Answer
Thus, the natural number is \( 121 \).

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