Question

If alpha and beta are the zeros of the polynomial 2x2+5x+k, find k such that (alpha)2+(beta)2+(alpha)*(beta)=24

Answer 1

k = -35.5

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 ...

• 5 = -2(α+β)
• k = 2αβ

__

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