If the sum of a positive number and its square is 240 , then find the number

Answer

To solve the problem where the sum of a positive number and its square is 240, we can follow these steps:

### Step-by-Step Solution:

1. **Let the number be \( x \)**:
- We define the unknown number as \( x \).

2. **Write the equation based on the problem statement**:
- According to the problem, the sum of the number and its square can be expressed as:
\[ x + x^2 = 240 \]

3. **Rearrange the equation**:
- To form a standard quadratic equation, we can rearrange the equation:
\[ x^2 + x - 240 = 0 \]

4. **Factor the quadratic equation**:
- We need to factor the quadratic equation \( x^2 + x - 240 = 0 \). We look for two numbers that multiply to \(-240\) and add to \(1\) (the coefficient of \(x\)).
- The numbers \(16\) and \(-15\) work because:
- \(16 \times (-15) = -240\)
- \(16 + (-15) = 1\)

5. **Rewrite the equation using the factors**:
- We can rewrite the quadratic as:
\[ (x + 16)(x - 15) = 0 \]

6. **Set each factor to zero**:
- Now we set each factor equal to zero:
\[ x + 16 = 0 \quad \text{or} \quad x - 15 = 0 \]

7. **Solve for \( x \)**:
- From \( x + 16 = 0 \):
\[ x = -16 \]
- From \( x - 15 = 0 \):
\[ x = 15 \]

8. **Determine the valid solution**:
- Since we are looking for a positive number, we discard \( x = -16 \) and accept:
\[ x = 15 \]

### Final Answer:
The positive number is \( \boxed{15} \).