The sum of a positive integer and its square is 2450. The positive integer is

Options

A. 45
B. 48
C. 49
D. 50

Answer

To solve the problem step by step, we need to find a positive integer \( A \) such that the sum of \( A \) and its square \( A^2 \) equals 2450.

### Step 1: Set up the equation
We start by expressing the problem mathematically. The sum of the integer \( A \) and its square \( A^2 \) can be written as:
\[ A + A^2 = 2450 \]

### Step 2: Rearrange the equation
Next, we rearrange this equation to bring all terms to one side:
\[ A^2 + A - 2450 = 0 \]

### Step 3: Factor the quadratic equation
Now, we need to factor the quadratic equation \( A^2 + A - 2450 = 0 \). We are looking for two numbers that multiply to \(-2450\) (the constant term) and add to \(1\) (the coefficient of \( A \)).

After testing pairs of factors, we find that:
\[ 49 \times (-50) = -2450 \quad \text{and} \quad 49 + (-50) = -1 \]
This means we can rewrite the equation as:
\[ A^2 + 50A - 49A - 2450 = 0 \]
Grouping the terms gives us:
\[ (A^2 + 50A) + (-49A - 2450) = 0 \]

### Step 4: Factor by grouping
Now we can factor by grouping:
\[ A(A + 50) - 49(A + 50) = 0 \]
Factoring out \( (A + 50) \):
\[ (A + 50)(A - 49) = 0 \]

### Step 5: Solve for \( A \)
Setting each factor equal to zero gives us:
1. \( A + 50 = 0 \) → \( A = -50 \) (not a positive integer)
2. \( A - 49 = 0 \) → \( A = 49 \) (this is a positive integer)

### Conclusion
The positive integer we are looking for is:
\[ \boxed{49} \]