If one zero of `p (y) = 4y ^(2) - 8 ky - 9` is negative of other, then find the value of K.

Answer

To solve the problem, we need to find the value of \( k \) such that one root of the polynomial \( p(y) = 4y^2 - 8ky - 9 \) is the negative of the other root.

### Step-by-Step Solution:

1. **Understanding the Roots**:
Let the two roots of the polynomial be \( \alpha \) and \( -\alpha \).

2. **Using the Sum of Roots**:
According to Vieta's formulas, the sum of the roots \( \alpha + (-\alpha) \) should equal to \( -\frac{b}{a} \), where \( b \) is the coefficient of \( y \) and \( a \) is the coefficient of \( y^2 \).
\[ \alpha + (-\alpha) = 0 \]
Thus, we have:
\[ 0 = -\frac{-8k}{4} \]

3. **Simplifying the Equation**:
Simplifying the right side:
\[ 0 = \frac{8k}{4} \]
This simplifies to:
\[ 0 = 2k \]

4. **Solving for \( k \)**:
To find \( k \), we set the equation \( 2k = 0 \):
\[ k = 0 \]

5. **Conclusion**:
The value of \( k \) is \( 0 \).

### Final Answer:
Thus, the value of \( k \) is \( 0 \).

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