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√5On squaring both the SIDES,⇒ α4 = 441 + 320 + 336√5⇒ α4 = 761 + 336√5Subtracting equation (2) form (1)α + β - α + β = 8 - 2√5⇒ β = 4 - √5⇒ β2 = 16 + 5 - 8√5⇒ β2 = 21 - 8√5On squaring both the sides,⇒ β4 = 441 + 320 - 336√5⇒ β4 = 761 - 336√5The 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

"> 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√5On squaring both the SIDES,⇒ α4 = 441 + 320 + 336√5⇒ α4 = 761 + 336√5Subtracting equation (2) form (1)α + β - α + β = 8 - 2√5⇒ β = 4 - √5⇒ β2 = 16 + 5 - 8√5⇒ β2 = 21 - 8√5On squaring both the sides,⇒ β4 = 441 + 320 - 336√5⇒ β4 = 761 - 336√5The 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

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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?

General Aptitude Algebra in General Aptitude 1 year ago

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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√5On squaring both the SIDES,⇒ α4 = 441 + 320 + 336√5⇒ α4 = 761 + 336√5Subtracting equation (2) form (1)α + β - α + β = 8 - 2√5⇒ β = 4 - √5⇒ β2 = 16 + 5 - 8√5⇒ β2 = 21 - 8√5On squaring both the sides,⇒ β4 = 441 + 320 - 336√5⇒ β4 = 761 - 336√5The 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

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