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 . 5 months 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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Posted on 08 Nov 2024, this text provides information on General Aptitude related to Algebra in General Aptitude. Please note that while accuracy is prioritized, the data presented might not be entirely correct or up-to-date. This information is offered for general knowledge and informational purposes only, and should not be considered as a substitute for professional advice.

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