Two year ago my age was `4(1)/(2)` times the age of my son . Six years ago, my age was twice the square of the age of my son. What is the present age of my son ?

Answer

To solve the problem, we will set up equations based on the information given in the question.

### Step 1: Define Variables
Let:
- \( x \) = present age of the father
- \( y \) = present age of the son

### Step 2: Set Up the First Equation
According to the first condition, two years ago, the father's age was \( 4 \frac{1}{2} \) times the age of his son.
This can be expressed as:
\[ x - 2 = 4.5(y - 2) \]
Converting \( 4.5 \) to a fraction:
\[ 4.5 = \frac{9}{2} \]
So the equation becomes:
\[ x - 2 = \frac{9}{2}(y - 2) \]

### Step 3: Simplify the First Equation
Expanding the equation:
\[ x - 2 = \frac{9}{2}y - \frac{9}{2} \cdot 2 \]
\[ x - 2 = \frac{9}{2}y - 9 \]
Now, rearranging gives:
\[ x = \frac{9}{2}y - 7 \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Equation
According to the second condition, six years ago, the father's age was twice the square of the age of his son. This can be expressed as:
\[ x - 6 = 2(y - 6)^2 \]

### Step 5: Simplify the Second Equation
Expanding the right side:
\[ x - 6 = 2(y^2 - 12y + 36) \]
\[ x - 6 = 2y^2 - 24y + 72 \]
Rearranging gives:
\[ x = 2y^2 - 24y + 78 \quad \text{(Equation 2)} \]

### Step 6: Substitute Equation 1 into Equation 2
Now we will substitute Equation 1 into Equation 2:
\[ \frac{9}{2}y - 7 = 2y^2 - 24y + 78 \]
Multiplying through by 2 to eliminate the fraction:
\[ 9y - 14 = 4y^2 - 48y + 156 \]

### Step 7: Rearranging the Equation
Rearranging gives:
\[ 4y^2 - 48y - 9y + 156 + 14 = 0 \]
\[ 4y^2 - 57y + 170 = 0 \]

### Step 8: Factor the Quadratic Equation
Now we will factor the quadratic equation:
\[ 4y^2 - 57y + 170 = 0 \]
Finding factors, we can write:
\[ (4y - 10)(y - 17) = 0 \]

### Step 9: Solve for \( y \)
Setting each factor to zero gives:
1. \( 4y - 10 = 0 \) → \( y = \frac{10}{4} = 2.5 \) (not valid since age cannot be a fraction)
2. \( y - 17 = 0 \) → \( y = 17 \)

### Conclusion
The present age of the son is \( \boxed{17} \).