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?

Question

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?

x² - 1522x + 14641 = 0
x² + 1921x + 14641 = 0
x²- 1764x + 14641 = 0
x²+ 2520x + 14641 = 0

LohaHridoy

6 views

2025-01-04 04:42

1 Answers

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AhmedNazir · Accepted Answer · 2022-09-04T04:29:57+06:00

Option 1 : 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

⇒ x² - 1522x + 14641 = 0