Quadratic EquationHard
Question
Let p and q be real numbers such that p ≠ 0, P3 ≠ -q. If α and β are nonzero complex numbers satisfying α + β = - p and α3 + β3 = q. then a quadratic equation having 
and 
as its roots is
and as its roots isOptions
A.(p3 + q)x2 - (p3 + 2q)x + (p3 + q) = 0
B.(p3 + q)x2 - (p3 - 2q)x + (p3 + q) = 0
C.(p3 - q)x2 - (5p3 - 2q)x + (p3 - q) = 0
D.(p3 - q)x2 - (5p3 + 2q)x + (p3 - q) = 0
Solution
α3 + β3 = q
⇒ (α + β)3 - 3αβ (α + β) = q
⇒ - p3 + 3pαβ = q ⇒ αβ =
x2 -
x2 -
x + 1 = 0
⇒ x2 -
x + 1 = 0
⇒ x2 -
x + 1 = 0
⇒ (p3 + q) x2 - (3p3 - 2p3 - 2q) x + (p3 + q) = 0
⇒ (p3 + q) x2 - (p3 - 2q) x + (p3 + q) = 0.
⇒ (α + β)3 - 3αβ (α + β) = q
⇒ - p3 + 3pαβ = q ⇒ αβ =
x2 -
x2 -
x + 1 = 0⇒ x2 -
x + 1 = 0⇒ x2 -
x + 1 = 0 ⇒ (p3 + q) x2 - (3p3 - 2p3 - 2q) x + (p3 + q) = 0
⇒ (p3 + q) x2 - (p3 - 2q) x + (p3 + q) = 0.
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