The age of a father is `25` years more than his son's age. The product of their ages is `84` in year. What will be son's age in years, after 10 years ?

Options

A. 3
B. 28
C. 13
D. 18

Explanation

To solve the problem step by step, we will denote the son's age as \( x \) years.

### Step 1: Set up the equations based on the problem statement.
- The father's age is 25 years more than the son's age, so we can express the father's age as \( x + 25 \).
- The product of their ages is given as 84. Therefore, we can write the equation:
\[ x \cdot (x + 25) = 84 \]

### Step 2: Expand the equation.
- Expanding the left side gives us:
\[ x^2 + 25x = 84 \]

### Step 3: Rearrange the equation to standard quadratic form.
- We can rearrange the equation to set it to zero:
\[ x^2 + 25x - 84 = 0 \]

### Step 4: Factor the quadratic equation.
- We need to factor the quadratic equation \( x^2 + 25x - 84 = 0 \). We look for two numbers that multiply to -84 and add to 25.
- The numbers that satisfy this are 28 and -3. Thus, we can write:
\[ (x + 28)(x - 3) = 0 \]

### Step 5: Solve for \( x \).
- Setting each factor to zero gives us:
\[ x + 28 = 0 \quad \text{or} \quad x - 3 = 0 \]
- From \( x + 28 = 0 \), we get \( x = -28 \) (not a valid age).
- From \( x - 3 = 0 \), we get \( x = 3 \).

### Step 6: Determine the son's age after 10 years.
- The son's current age is \( 3 \) years. After 10 years, his age will be:
\[ 3 + 10 = 13 \text{ years} \]

### Final Answer:
- The son's age after 10 years will be **13 years**.

---