Condition for Reciprocal Zeroes of a Quadratic Polynomial

Video Explanation

Question

If one zero of the polynomial

\[ f(x) = (k^2 + 4)x^2 + 13x + 4k \]

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

Solution

Step 1: Use the Condition for Reciprocal Zeroes

If the zeroes of a quadratic polynomial are reciprocals of each other, then

\[ \alpha\beta = 1 \]

Step 2: Write Product of Zeroes Using Coefficients

For the quadratic polynomial \[ ax^2 + bx + c, \]

\[ \alpha\beta = \frac{c}{a} \]

Here,

\[ a = k^2 + 4, \quad c = 4k \]

So,

\[ \frac{4k}{k^2 + 4} = 1 \]

Step 3: Solve for \(k\)

\[ 4k = k^2 + 4 \]

\[ k^2 – 4k + 4 = 0 \]

\[ (k – 2)^2 = 0 \]

\[ \Rightarrow k = 2 \]

Conclusion

The value of \(k\) for which one zero is the reciprocal of the other is

\[ \boxed{k = 2} \]