Question

If α and β are the zeros of the polynomial p(x) = 2x² + 5x + k satisfying the relation α² + β² + αβ = `21/4` then find the value of k.

Answer 1

Given: α and β are zeros of p(x) = 2x² + 5x + k and α² + β² + αβ = `21/4`.

Step-wise calculation:

1. Sum and product of roots:

`α + β = -5/2` 

`αβ = k/2`

2. Express α² + β² + αβ = (α + β)² – αβ.

3. Substitute: `(-5/2)^2 - (k/2) = 21/4`

`25/4 - k/2 = 21/4`

4. Solve: `k/2 = (25/4 - 21/4) = 1`

k = 2