Question

If α and β (≠ 0) are the roots of the quadratic equation x² + αx - β = 0,  then the quadratic expression  -x²  + αx + β  where x ∈ R has 

a. Least value - 1/4 

b. Least value - 9/4 

c. Greatest value 1/4 

d. Greatest value 9/4

Answer 1

Correct option d. Greatest value 9/4

Explanation:

α + β = - α, αβ = - β  ⇒ αβ + β = 0

⇒ (α + 1)β = 0 ⇒ α = - 1 (β ≠ 0)

⇒ 2α + β = 0 

⇒ β = 2

∴ -x² + x + = - x² - x + 2

Greatest value = - (1 + 8)/-4 = 9/4