Condition on Zeros of a Quadratic Polynomial

Video Explanation

Question

If one of the zeroes of the quadratic polynomial

\[ f(x) = 4x^2 – 8kx – 9 \]

is the negative of the other, find the value of \(k\).

Solution

Step 1: Use the Given Condition

Let the zeroes be \( \alpha \) and \( \beta \).

Given that one zero is the negative of the other:

\[ \beta = -\alpha \]

So,

\[ \alpha + \beta = 0 \]

Step 2: Use Relation Between Zeros and Coefficients

For a quadratic polynomial \( ax^2 + bx + c \),

\[ \alpha + \beta = -\frac{b}{a} \]

Here,

\[ a = 4,\quad b = -8k \]

\[ \alpha + \beta = -\frac{-8k}{4} = 2k \]

Step 3: Apply the Condition

Since \( \alpha + \beta = 0 \),

\[ 2k = 0 \]

\[ k = 0 \]

Conclusion

The value of \(k\) is:

\[ \boxed{0} \]

\[ \therefore \quad k = 0. \]