Answer:
k = -35.5
Step-by-step explanation:
When α and β are zeros of the polynomial 2x² +5x +k, it can be written in factored form as ...
2x² +5x +k = 2(x -α)(x -β)
When the factored form is expanded, it becomes ...
= 2x² -2(α+β)x +2αβ
Comparing this to the original, we see that ...
__
Then the given expression can be used to find k:
α +β = -5/2 . . . . . solved the first relation for α+β
(α +β)² = (α² +β² +αβ) +αβ = 24 +αβ
(-5/2)² = 24 +αβ . . . . . . . use -5/2 for α+β
αβ = 25/4 -24 = -17.75 . . . . . solve for αβ
Then the value of k is ...
k = 2αβ = 2(-17.75)
k = -35.5
