If one zero of the polynomial f(x) = (k² + 4)x² + 13x + 4k is the reciprocal of the other, find the value of k

Video Explanation

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Given

f(x) = (k² + 4)x² + 13x + 4k

Let the zeroes of the polynomial be α and β.

It is given that one zero is the reciprocal of the other.

To Find

The value of k.

Concept Used

If one zero is the reciprocal of the other, then:

αβ = 1

For a quadratic polynomial ax² + bx + c:

αβ = c/a

Solution

Comparing f(x) = (k² + 4)x² + 13x + 4k with ax² + bx + c,

a = k² + 4,   b = 13,   c = 4k

Step 1: Use the Condition αβ = 1

αβ = c/a

⇒ 4k / (k² + 4) = 1

Step 2: Solve for k

4k = k² + 4

k² − 4k + 4 = 0

(k − 2)² = 0

⇒ k = 2

Final Answer

k = 2

Conclusion

Hence, if one zero of the polynomial f(x) = (k² + 4)x² + 13x + 4k is the reciprocal of the other, then the value of k is 2.