If the cubic `2x^3=9x^2+12 x+k=0` has two equal roots then minimum value of `|k|` is______.

Answer

5

Explanation

We have `2x^(3)-9x^(2)+12x +k =0`
Let the roots are `alpha,alpha, beta`
`2alpha+beta=(9)/(2)" "(1)`
`alpha + 2alphabet=(12)/(2)=6" "(2)`
are `alpha^(2)beta=-(k)/(2)" "(3)`
putting `beta = ((9)/(2)-2alpha)` from (1) in (2), we have
`alpha^(2)+2alpha((9)/(2)-2alpha)=6`
or `alpha^(2)+9alpha-4alpha^(2)=6`
or `3alpha^(2)-9alpha+6=0`
or `alpha^(2)-3alpha+2=0`
or `(alpha-2)(alpha-1)=0rArralpha=2 or1`
`if alpha=2, "then "alpha=(1)/(2),`
`if alpha=1, " then " beta=(5)/(2)`
`therefore k=-2(2^(2))(1)/(2)=-4`
`or k=-2 (1^(2))((5)/(2))=-5`