The sum of ages of a boy and his brother is 25 years, and the product of their ages in years is 126. Find their ages.

Answer

To solve the problem, we need to find the ages of a boy and his brother given that their ages sum up to 25 years and their ages multiply to 126 years. Let's denote the age of the boy as \( x \) and the age of his brother as \( y \).

### Step-by-step Solution:

1. **Set up the equations based on the problem statement**:
- From the problem, we have two equations:
\[ x + y = 25 \quad \text{(1)} \]
\[ xy = 126 \quad \text{(2)} \]

2. **Express one variable in terms of the other**:
- From equation (1), we can express \( y \) in terms of \( x \):
\[ y = 25 - x \quad \text{(3)} \]

3. **Substitute equation (3) into equation (2)**:
- Substitute \( y \) from equation (3) into equation (2):
\[ x(25 - x) = 126 \]
- This simplifies to:
\[ 25x - x^2 = 126 \]

4. **Rearrange the equation**:
- Rearranging gives us a standard form of a quadratic equation:
\[ x^2 - 25x + 126 = 0 \quad \text{(4)} \]

5. **Factor the quadratic equation**:
- We need to factor equation (4). We look for two numbers that multiply to 126 and add up to 25. The numbers are 18 and 7.
- Thus, we can factor the equation as:
\[ (x - 18)(x - 7) = 0 \]

6. **Solve for \( x \)**:
- Setting each factor to zero gives us:
\[ x - 18 = 0 \quad \Rightarrow \quad x = 18 \]
\[ x - 7 = 0 \quad \Rightarrow \quad x = 7 \]

7. **Find the corresponding values of \( y \)**:
- For \( x = 18 \):
\[ y = 25 - 18 = 7 \]
- For \( x = 7 \):
\[ y = 25 - 7 = 18 \]

8. **Conclusion**:
- The ages of the boy and his brother are 18 years and 7 years.

### Final Answer:
The ages of the boy and his brother are 18 years and 7 years.