α and β are the roots of quadratic equation. If α + β = 8 and α – β = 2√5, then which of the following equation will have roots α4 and β4?
Show answer & explanation
Correct answer: A
Explanation
The correct answer is option 1 i.e. x2 – 1522x + 14641 = 0
α + β = 8 .....(1)
α - β = 2√5 .....(2)
By solving euqation (1) and (2), we get
α = 4 + √5, β = 4 - √5
α + β = 8
αβ = 11
α2 + β2 = (α + β) - 2αβ = 82 - 2 × 11 = 42
α4 + β4 = (α2 + β2)2 - 2α2β2 = 422 - 2 × 112 = 1764 - 242 = 1522
α4β4 = 114 = 14641
Equation
x2 - (α4 + β4)x + (α4β4) = 0
x2 – 1522x + 14641 = 0