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2x^(2) + 3x - alpha - 0 " has roots "-2 ...

`2x^(2) + 3x - alpha - 0 " has roots "-2 and beta " while the equation "x^(2) - 3mx + 2m^(2) = 0 " has both roots positive, where " alpha gt 0 and beta gt 0.`
It is given that the roots of the equation `x^(2) - 4x - log_(3) P=0` are real . For this, the minimum value of P is

A

`1/27`

B

`1/64`

C

`1/81`

D

1

Text Solution

Verified by Experts

The correct Answer is:
C
`x^4x - log_(3) P=0`
We know , roots are real if dis criminant is greater than or equal to 0.
i.e., ` b^(2) - 4ac ge 0 rArr b^(2) ge 4ac`
In the given equation , ` a=1, b=-4, c=-log_(3) P.`
` :. b^(2) ge 4ac rArr (-4)^(2) ge 4(1) (-log_(3)P)`
`rArr 16 ge -4log_(3) P`
` rArr 4 ge -log_(3)P`
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