If one zero of the polynomial `(a^(2)+9)x^(2)+13x+6a` is reciprocal of the other, find the value of a.

Explanation

To solve the problem, we need to find the value of \( a \) such that one zero of the polynomial \( (a^2 + 9)x^2 + 13x + 6a \) is the reciprocal of the other.

### Step-by-step Solution:

1. **Identify the Polynomial**:
The polynomial is given as:
\[ P(x) = (a^2 + 9)x^2 + 13x + 6a \]

2. **Use the Property of Reciprocals**:
If one root (zero) of the polynomial is \( \alpha \), then the other root is \( \frac{1}{\alpha} \). For a quadratic polynomial \( ax^2 + bx + c \), the product of the roots is given by:
\[ \text{Product of roots} = \frac{c}{a} \]
Here, \( a = a^2 + 9 \), \( b = 13 \), and \( c = 6a \).

3. **Set Up the Equation**:
Since the roots are \( \alpha \) and \( \frac{1}{\alpha} \), their product is:
\[ \alpha \cdot \frac{1}{\alpha} = 1 \]
Therefore, we can equate this to the product of the roots from the polynomial:
\[ 1 = \frac{6a}{a^2 + 9} \]

4. **Cross-Multiply**:
Cross-multiplying gives:
\[ a^2 + 9 = 6a \]

5. **Rearrange the Equation**:
Rearranging the equation leads to:
\[ a^2 - 6a + 9 = 0 \]

6. **Factor the Quadratic**:
The quadratic can be factored as:
\[ (a - 3)(a - 3) = 0 \]
This simplifies to:
\[ (a - 3)^2 = 0 \]

7. **Solve for \( a \)**:
Setting the factor equal to zero gives:
\[ a - 3 = 0 \implies a = 3 \]

### Conclusion:
The value of \( a \) is \( 3 \).

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