Condition for Reciprocal Zeroes of a Quadratic Polynomial

Video Explanation

Question

If one root of the polynomial

\[ f(x) = 5x^2 + 13x + k \]

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

Solution

Step 1: Use the Condition for Reciprocal Roots

If the roots \( \alpha \) and \( \beta \) are reciprocals of each other, then

\[ \alpha \beta = 1 \]

Step 2: Use the Product of Roots Formula

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

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

Here,

\[ a = 5, \quad c = k \]

So,

\[ \frac{k}{5} = 1 \]

Step 3: Solve for \(k\)

\[ k = 5 \]

Conclusion

The value of \(k\) is:

\[ \boxed{5} \]