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If a, b. c in R and 3b^(2)-8ac lt 0, the...

If `a,` b`. `c in R` and `3b^(2)-8ac lt 0`, then the equation `ax^(4)+bx^(3)+cx^(2)+5x-7=0` has

A

all real roots

B

all imaginary roots

C

exactly two real and two imaginary roots

D

none

Text Solution

Verified by Experts

The correct Answer is:
C
`(c )` `f(x)=ax^(4)+bx^(3)+cx^(2)+5x-7`
`f'(x)=4ax^(3)+3bx^(2)+2cx+5`
`f''(x)=12ax^(2)+6bx+2c=0`
`implies "disc"=36b^(2)-4(12a)(2c)`
`=12(3b^(2)-8ac) lt 0`
`(:' 3b^(2) lt 8ac)`
`f'(x)=0` has only one real root
`implies f(x)=0` has exactly two real roots.
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