If x is a positive integers and `x+12=x^(2)`, what is the value of x?

Options

A. `2`
B. `4`
C. `6`
D. `12`

Answer

To solve the equation \( x + 12 = x^2 \) for positive integers \( x \), we can follow these steps:

### Step 1: Rearrange the equation
Start by moving all terms to one side of the equation:
\[ x^2 - x - 12 = 0 \]

### Step 2: Identify the quadratic equation
The equation \( x^2 - x - 12 = 0 \) is a standard quadratic equation in the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -1 \), and \( c = -12 \).

### Step 3: Factor the quadratic equation
We need to factor the quadratic equation. We are looking for two numbers that multiply to \( -12 \) (the constant term) and add up to \( -1 \) (the coefficient of \( x \)). The numbers \( -4 \) and \( 3 \) satisfy these conditions:
\[ x^2 - 4x + 3x - 12 = 0 \]
This can be grouped as:
\[ (x - 4)(x + 3) = 0 \]

### Step 4: Solve for \( x \)
Setting each factor to zero gives us the possible solutions:
1. \( x - 4 = 0 \) → \( x = 4 \)
2. \( x + 3 = 0 \) → \( x = -3 \)

### Step 5: Determine the valid solution
Since we are looking for positive integers, we discard \( x = -3 \). Thus, the only valid solution is:
\[ x = 4 \]

### Final Answer
The value of \( x \) is \( 4 \).

---