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 step by step, we will follow these steps:

### Step 1: Define Variables
Let the age of the boy be \( x \) years and the age of his brother be \( y \) years.

### Step 2: Set Up the Equations
According to the problem, we have two equations:
1. The sum of their ages:
\[ x + y = 25 \quad \text{(Equation 1)} \]
2. The product of their ages:
\[ x \cdot y = 126 \quad \text{(Equation 2)} \]

### Step 3: Express One Variable in Terms of the Other
From Equation 1, we can express \( y \) in terms of \( x \):
\[ y = 25 - x \]

### Step 4: Substitute into the Second Equation
Now, substitute \( y \) in Equation 2:
\[ x \cdot (25 - x) = 126 \]
Expanding this gives:
\[ 25x - x^2 = 126 \]

### Step 5: Rearrange the Equation
Rearranging the equation to standard quadratic form:
\[ -x^2 + 25x - 126 = 0 \]
Multiplying through by -1 to make the leading coefficient positive:
\[ x^2 - 25x + 126 = 0 \quad \text{(Equation 3)} \]

### Step 6: Factor the Quadratic Equation
Now we need to factor Equation 3. We are looking for two numbers that multiply to \( 126 \) and add up to \( 25 \). The numbers are \( 18 \) and \( 7 \):
\[ (x - 18)(x - 7) = 0 \]

### Step 7: Solve for \( x \)
Setting each factor to zero gives us:
1. \( x - 18 = 0 \) → \( x = 18 \)
2. \( x - 7 = 0 \) → \( x = 7 \)

### Step 8: Find Corresponding Values of \( y \)
Now, we can find the corresponding values of \( y \) for each value of \( x \):
1. If \( x = 18 \):
\[ y = 25 - 18 = 7 \]
2. If \( x = 7 \):
\[ y = 25 - 7 = 18 \]

### Conclusion
Thus, the ages of the boy and his brother are:
- One possibility: The boy is \( 18 \) years old and his brother is \( 7 \) years old.
- The other possibility: The boy is \( 7 \) years old and his brother is \( 18 \) years old.