If (α + √β ) and (α -√β ) are the roots of the equation where x2 + px + q = 0, where α, β p and q are real, then the roots of the equation (p2 - 4q) (p2x2 + 4px) - 16q = 0 are

1) [(1 / ɑ) + (1 / √β)] and [(1 / ɑ) – (1 / √β)]

2) [(1 / √ɑ) + (1 / β)] and [(1 / √ɑ) – (1 / β)]

3) [(1 / √ɑ) + (1 / √β)] and [(1 / √ɑ) – (1 / √β)]

4) [(√ɑ) + (√β)] and [(√ɑ) – (√β)]

Solution: (1) [(1 / ɑ) + (1 / √β)] and [(1 / ɑ) – (1 / √β)]

(α + √β ) + (α – √β) = – p

The equation x2 + px + q = 0

α + √β + α – √β = – p; α2 – β = q

2α = – p; α2 – β = q

p2 / 4 – β = q

β = [p2 – 4q] / 4

[p2 – 4q] [p2x2 + 4px] – 16q = 0

4β (4α2x2 – 8αx) – 16 (α2 – β) = 0

β (4α2x2 – 8αx) – 4 (α2 – β) = 0

[(α √β x) – √β]2 = α2

α √β x – √β = ± α

x = [(1 / ɑ) + (1 / √β)] and [(1 / ɑ) – (1 / √β)]

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