α and β are the roots of the quadratic equation. If α + β = 8 and α - β = 2√5, then which of the following equation will have roots α4 and β4?

α and β are the roots of the quadratic equation. If α + β = 8 and α - β = 2√5, then which of the following equation will have roots α4 and β4? Correct Answer x² - 1522x + 14641 = 0

α + β = 8      ----(1)

α - β = 2√5      ----(2)

Adding equation (1) and (2)

⇒ α + β + α - β = 8 + 2√5

⇒ 2α = 8 + 2√5

⇒ α = 4 + √5

⇒ α² = 16 + 5 + 8√5

⇒ α² = 21 + 8√5

On squaring both the sides,

⇒ α⁴ = 441 + 320 + 336√5

⇒ α⁴ = 761 + 336√5

Subtracting equation (2) form (1)

α + β - α + β = 8 - 2√5

⇒ β = 4 - √5

⇒ β² = 16 + 5 - 8√5

⇒ β² = 21 - 8√5

On squaring both the sides,

⇒ β⁴ = 441 + 320 - 336√5

⇒ β⁴ = 761 - 336√5

The required equation format, x² - (sum of roots)x + (product of roots) = 0

⇒ x² - (α⁴ + β⁴)x + αβ = 0

⇒ x² - (761 + 336√5 +761 - 336√5)x + (761 + 336√5)(761 - 336√5) = 0

⇒ x² - 1522x + (761 + 336√5)(761 - 336√5) = 0